Number Fields

• B1.9

  Number Fields

  Part II, 2001

  Let $K=\mathbf{Q}(\alpha)$ be a number field, where $\alpha \in \mathcal{O}_{K}$. Let $f$ be the (normalized) minimal polynomial of $\alpha$ over $Q$. Show that the discriminant $\operatorname{disc}(f)$ of $f$ is equal to $\left(\mathcal{O}_{K}: \mathbf{Z}[\alpha]\right)^{2} D_{K}$.

  Show that $f(x)=x^{3}+5 x^{2}-19$ is irreducible over Q. Determine $\operatorname{disc}(f)$ and the ring of algebraic integers $\mathcal{O}_{K}$ of $K=\mathbf{Q}(\alpha)$, where $\alpha \in \mathbf{C}$ is a root of $f$.

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• B2.9

  Number Fields

  Part II, 2001

  Determine the ideal class group of $\mathbf{Q}(\sqrt{-11})$.

  Find all solutions of the diophantine equation

  $$y^{2}+11=x^{3} \quad(x, y \in \mathbf{Z})$$

  [Minkowski's bound is $n ! n^{-n}(4 / \pi)^{r_{2}}\left|D_{k}\right|^{1 / 2}$.]

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• B4.6

  Number Fields

  Part II, 2001

  For a prime number $p>2$, set $\zeta=e^{2 \pi i / p}, K=\mathbf{Q}(\zeta)$ and $K^{+}=\mathbf{Q}\left(\zeta+\zeta^{-1}\right)$.

  (a) Show that the (normalized) minimal polynomial of $\zeta-1$ over $\mathbf{Q}$ is equal to

  $$f(x)=\frac{(x+1)^{p}-1}{x} .$$

  (b) Determine the degrees $[K: \mathbf{Q}]$ and $\left[K^{+}: \mathbf{Q}\right]$.

  (c) Show that

  $$\prod_{j=1}^{p-1}\left(1-\zeta^{j}\right)=p$$

  (d) Show that $\operatorname{disc}(f)=(-1)^{\frac{p-1}{2}} p^{p-2}$.

  (e) Show that $K$ contains $\mathbf{Q}\left(\sqrt{p^{*}}\right)$, where $p^{*}=(-1)^{\frac{p-1}{2}} p$.

  (f) If $j, k \in \mathbf{Z}$ are not divisible by $p$, show that $\frac{1-\zeta^{j}}{1-\zeta^{k}}$ lies in $\mathcal{O}_{K}^{*}$.

  (g) Show that the ideal $(p)=p \mathcal{O}_{K}$ is equal to $(1-\zeta)^{p-1}$.

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• B2.9

  Number Fields

  Part II, 2002

  Let $K=\mathbb{Q}(\sqrt{35})$. By Dedekind's theorem, or otherwise, show that the ideal equations

  $$2=[2, \omega]^{2}, \quad 5=[5, \omega]^{2}, \quad[\omega]=[2, \omega][5, \omega]$$

  hold in $K$, where $\omega=5+\sqrt{35}$. Deduce that $K$ has class number 2 .

  Verify that $1+\omega$ is the fundamental unit in $K$. Hence show that the complete solution in integers $x, y$ of the equation $x^{2}-35 y^{2}=-10$ is given by

  $$x+\sqrt{35} y=\pm \omega(1+\omega)^{n} \quad(n=0, \pm 1, \pm 2, \ldots) .$$

  Calculate the particular solution $x, y$ for $n=1$.

  [It can be assumed that the Minkowski constant for $K$ is $\frac{1}{2}$.]

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• B4.6

  Number Fields

  Part II, 2002

  Write an essay on one the following topics.

  (i) Dirichlet's unit theorem and the Pell equation.

  (ii) Ideals and the fundamental theorem of arithmetic.

  (iii) Dedekind's theorem and the factorisation of primes. (You should treat explicitly either the case of quadratic fields or that of the cyclotomic field.)

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• B1.9

  Number Fields

  Part II, 2002

  Explain what is meant by an integral basis $\omega_{1}, \ldots, \omega_{n}$ of a number field $K$. Give an expression for the discriminant of $K$ in terms of the traces of the $\omega_{i} \omega_{j}$.

  Let $K=\mathbb{Q}(i, \sqrt{2})$. By computing the traces $T_{K / k}(\theta)$, where $k$ runs through the three quadratic subfields of $K$, show that the algebraic integers $\theta$ in $K$ have the form $\frac{1}{2}(\alpha+\beta \sqrt{2})$, where $\alpha=a+i b$ and $\beta=c+i d$ are Gaussian integers. By further computing the norm $N_{K / k}(\theta)$, where $k=\mathbb{Q}(\sqrt{2})$, show that $a$ and $b$ are even and that $c \equiv d(\bmod 2)$. Hence prove that an integral basis for $K$ is $1, i, \sqrt{2}, \frac{1}{2}(1+i) \sqrt{2}$.

  Calculate the discriminant of $K$.

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• B2.9

  Number Fields

  Part II, 2003

  By Dedekind's theorem, or otherwise, factorise $2,3,5$ and 7 into prime ideals in the field $K=\mathbb{Q}(\sqrt{-34})$. Show that the ideal equations

  $$[\omega]=[5, \omega][7, \omega], \quad[\omega+3]=[2, \omega+3][5, \omega+3]^{2}$$

  hold in $K$, where $\omega=1+\sqrt{-34}$. Hence, prove that the ideal class group of $K$ is cyclic of order $4 .$

  [It may be assumed that the Minkowski constant for $K$ is $2 / \pi$.]

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• B4.6

  Number Fields

  Part II, 2003

  Write an essay on the Dirichlet unit theorem with particular reference to quadratic fields.

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• B1.9

  Number Fields

  Part II, 2003

  Let $K=\mathbb{Q}(\alpha)$, where $\alpha=\sqrt[3]{10}$, and let $\mathcal{O}_{K}$ be the ring of algebraic integers of $K$. Show that the field polynomial of $r+s \alpha$, with $r$ and $s$ rational, is $(x-r)^{3}-10 s^{3}$.

  Let $\beta=\frac{1}{3}\left(\alpha^{2}+\alpha+1\right)$. By verifying that $\beta=3 /(\alpha-1)$ and determining the field polynomial, or otherwise, show that $\beta$ is in $\mathcal{O}_{K}$.

  By computing the traces of $\theta, \alpha \theta, \alpha^{2} \theta$, show that the elements of $\mathcal{O}_{K}$ have the form

  $$\theta=\frac{1}{3}\left(l+\frac{1}{10} m \alpha+\frac{1}{10} n \alpha^{2}\right)$$

  where $l, m, n$ are integers. By further computing the norm of $\frac{1}{10} \alpha(m+n \alpha)$, show that $\theta$ can be expressed as $\frac{1}{3}(u+v \alpha)+w \beta$ with $u, v, w$ integers. Deduce that $1, \alpha, \beta$ form an integral basis for $K$.

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• B1.9

  Number Fields

  Part II, 2004

  Let $K=\mathbb{Q}(\theta)$, where $\theta$ is a root of $X^{3}-4 X+1$. Prove that $K$ has degree 3 over $\mathbb{Q}$, and admits three distinct embeddings in $\mathbb{R}$. Find the discriminant of $K$ and determine the ring of integers $\mathcal{O}$ of $K$. Factorise $2 \mathcal{O}$ and $3 \mathcal{O}$ into a product of prime ideals.

  Using Minkowski's bound, show that $K$ has class number 1 provided all prime ideals in $\mathcal{O}$ dividing 2 and 3 are principal. Hence prove that $K$ has class number $1 .$

  [You may assume that the discriminant of $X^{3}+a X+b$ is $-4 a^{3}-27 b^{2}$.]

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• B2.9

  Number Fields

  Part II, 2004

  Let $m$ be an integer greater than 1 and let $\zeta_{m}$ denote a primitive $m$-th root of unity in $\mathbb{C}$. Let $\mathcal{O}$ be the ring of integers of $\mathbb{Q}\left(\zeta_{m}\right)$. If $p$ is a prime number with $(p, m)=1$, outline the proof that

  $$p \mathcal{O}=\wp_{1} \cdots \wp_{r},$$

  where the $\wp_{i}$ are distinct prime ideals of $\mathcal{O}$, and $r=\varphi(m) / f$ with $f$ the least integer $\geqslant 1$ such that $p^{f} \equiv 1 \bmod m$. [Here $\varphi(m)$ is the Euler $\varphi$-function of $\left.m\right]$.

  Determine the factorisations of $2,3,5$ and 11 in $\mathbb{Q}\left(\zeta_{5}\right)$. For each integer $n \geqslant 1$, prove that, in the ring of integers of $\mathbb{Q}\left(\zeta_{5^{n}}\right)$, there is a unique prime ideal dividing 2 , and a unique prime ideal dividing 3 .

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• B4.6

  Number Fields

  Part II, 2004

  Let $K$ be a finite extension of $\mathbb{Q}$, and $\mathcal{O}$ the ring of integers of $K$. Write an essay outlining the proof that every non-zero ideal of $\mathcal{O}$ can be written as a product of non-zero prime ideals, and that this factorisation is unique up to the order of the factors.

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• 1.II.20G

  Number Fields

  Part II, 2005

  Let $K=\mathbb{Q}(\sqrt{2}, \sqrt{p})$ where $p$ is a prime with $p \equiv 3 \operatorname{(\operatorname {mod}4)\text {.Bycomputingthe}}$ relative traces $\operatorname{Tr}_{K / k}(\theta)$ where $k$ runs through the three quadratic subfields of $K$, show that the algebraic integers $\theta$ in $K$ have the form

  $$\theta=\frac{1}{2}(a+b \sqrt{p})+\frac{1}{2}(c+d \sqrt{p}) \sqrt{2}$$

  where $a, b, c, d$ are rational integers. By further computing the relative norm $\mathrm{N}_{K / k}(\theta)$ where $k=\mathbb{Q}(\sqrt{2})$, show that 4 divides

  $$a^{2}+p b^{2}-2\left(c^{2}+p d^{2}\right) \quad \text { and } \quad 2(a b-2 c d)$$

  Deduce that $a$ and $b$ are even and $c \equiv d(\bmod 2)$. Hence verify that an integral basis for $K$ is

  $$1, \quad \sqrt{2}, \quad \sqrt{p}, \quad \frac{1}{2}(1+\sqrt{p}) \sqrt{2} .$$

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• 2.II.20G

  Number Fields

  Part II, 2005

  Show that $\varepsilon=(3+\sqrt{7}) /(3-\sqrt{7})$ is a unit in $k=\mathbb{Q}(\sqrt{7})$. Show further that 2 is the square of the principal ideal in $k$ generated by $3+\sqrt{7}$.

  Assuming that the Minkowski constant for $k$ is $\frac{1}{2}$, deduce that $k$ has class number 1 .

  Assuming further that $\varepsilon$ is the fundamental unit in $k$, show that the complete solution in integers $x, y$ of the equation $x^{2}-7 y^{2}=2$ is given by

  $$x+\sqrt{7} y=\pm \varepsilon^{n}(3+\sqrt{7}) \quad(n=0, \pm 1, \pm 2, \ldots) .$$

  Calculate the particular solution in positive integers $x, y$ when $n=1$

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• 4.II.20G

  Number Fields

  Part II, 2005

  State Dedekind's theorem on the factorisation of rational primes into prime ideals.

  A rational prime is said to ramify totally in a field with degree $n$ if it is the $n$-th power of a prime ideal in the field. Show that, in the quadratic field $\mathbb{Q}(\sqrt{d})$ with $d$ a squarefree integer, a rational prime ramifies totally if and only if it divides the discriminant of the field.

  Verify that the same holds in the cyclotomic field $\mathbb{Q}(\zeta)$, where $\zeta=e^{2 \pi i / q}$ with $q$ an odd prime, and also in the cubic field $\mathbb{Q}(\sqrt[3]{2})$.

  $[$ The cases $d \equiv 2,3(\bmod 4)$ and $d \equiv 1(\bmod 4)$ for the quadratic field should be carefully distinguished. It can be assumed that $\mathbb{Q}(\zeta)$ has a basis $1, \zeta, \ldots, \zeta^{q-2}$ and discriminant $(-1)^{(q-1) / 2} q^{q-1}$, and that $\mathbb{Q}(\sqrt[3]{2})$ has a basis $1, \sqrt[3]{2},(\sqrt[3]{2})^{2}$ and discriminant $\left.-108 .\right]$

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• 1.II.20G

  Number Fields

  Part II, 2006

  Let $\alpha, \beta, \gamma$ denote the zeros of the polynomial $x^{3}-n x-1$, where $n$ is an integer. The discriminant of the polynomial is defined as

  $$\Delta=\Delta\left(1, \alpha, \alpha^{2}\right)=(\alpha-\beta)^{2}(\beta-\gamma)^{2}(\gamma-\alpha)^{2}$$

  Prove that, if $\Delta$ is square-free, then $1, \alpha, \alpha^{2}$ is an integral basis for $k=\mathbb{Q}(\alpha)$.

  By verifying that

  $$\alpha(\alpha-\beta)(\alpha-\gamma)=2 n \alpha+3$$

  and further that the field norm of the expression on the left is $-\Delta$, or otherwise, show that $\Delta=4 n^{3}-27$. Hence prove that, when $n=1$ and $n=2$, an integral basis for $k$ is $1, \alpha, \alpha^{2}$.

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• 2.II.20G

  Number Fields

  Part II, 2006

  Let $K=\mathbb{Q}(\sqrt{26})$ and let $\varepsilon=5+\sqrt{26}$. By Dedekind's theorem, or otherwise, show that the ideal equations

  $$2=[2, \varepsilon+1]^{2}, \quad 5=[5, \varepsilon+1][5, \varepsilon-1], \quad[\varepsilon+1]=[2, \varepsilon+1][5, \varepsilon+1]$$

  hold in $K$. Deduce that $K$ has class number 2 .

  Show that $\varepsilon$ is the fundamental unit in $K$. Hence verify that all solutions in integers $x, y$ of the equation $x^{2}-26 y^{2}=\pm 10$ are given by

  $$x+\sqrt{26} y=\pm \varepsilon^{n}(\varepsilon \pm 1) \quad(n=0, \pm 1, \pm 2, \ldots) .$$

  [It may be assumed that the Minkowski constant for $K$ is $\frac{1}{2}$.]

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• 4.II.20G

  Number Fields

  Part II, 2006

  Let $\zeta=e^{2 \pi i / 5}$ and let $K=\mathbb{Q}(\zeta)$. Show that the discriminant of $K$ is 125 . Hence prove that the ideals in $K$ are all principal.

  Verify that $\left(1-\zeta^{n}\right) /(1-\zeta)$ is a unit in $K$ for each integer $n$ with $1 \leqslant n \leqslant 4$. Deduce that $5 /(1-\zeta)^{4}$ is a unit in $K$. Hence show that the ideal $[1-\zeta]$ is prime and totally ramified in $K$. Indicate briefly why there are no other ramified prime ideals in $K$.

  [It can be assumed that $\zeta, \zeta^{2}, \zeta^{3}, \zeta^{4}$ is an integral basis for $K$ and that the Minkowski constant for $K$ is $3 /\left(2 \pi^{2}\right)$.]

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• 1.II.20H

  Number Fields

  Part II, 2007

  Let $K=\mathbb{Q}(\sqrt{-26})$.

  (a) Show that $\mathcal{O}_{K}=\mathbb{Z}[\sqrt{-26}]$ and that the discriminant $d_{K}$ is equal to $-104$.

  (b) Show that 2 ramifies in $\mathcal{O}_{K}$ by showing that $[2]=\mathfrak{p}_{2}^{2}$, and that $\mathfrak{p}_{2}$ is not a principal ideal. Show further that $[3]=\mathfrak{p}_{3} \overline{\mathfrak{p}}_{3}$ with $\mathfrak{p}_{3}=[3,1-\sqrt{-26}]$. Deduce that neither $\mathfrak{p}_{3}$ nor $\mathfrak{p}_{3}^{2}$ is a principal ideal, but $\mathfrak{p}_{3}^{3}=[1-\sqrt{-26}]$.

  (c) Show that 5 splits in $\mathcal{O}_{K}$ by showing that $[5]=\mathfrak{p}_{5} \overline{\mathfrak{p}}_{5}$, and that

  $$N_{K / \mathbb{Q}}(2+\sqrt{-26})=30$$

  Deduce that $\mathfrak{p}_{2} \mathfrak{p}_{3} \mathfrak{p}_{5}$ has trivial class in the ideal class group of $K$. Conclude that the ideal class group of $K$ is cyclic of order six.

  [You may use the fact that $\left.\frac{2}{\pi} \sqrt{104} \approx 6.492 .\right]$

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• 2.II.20H

  Number Fields

  Part II, 2007

  Let $K=\mathbb{Q}(\sqrt{10})$ and put $\varepsilon=3+\sqrt{10}$.

  (a) Show that 2,3 and $\varepsilon+1$ are irreducible elements in $\mathcal{O}_{K}$. Deduce from the equation

  $$6=2 \cdot 3=(\varepsilon+1)(\bar{\varepsilon}+1)$$

  that $\mathcal{O}_{K}$ is not a principal ideal domain.

  (b) Put $\mathfrak{p}_{2}=[2, \varepsilon+1]$ and $\mathfrak{p}_{3}=[3, \varepsilon+1]$. Show that

  $$[2]=\mathfrak{p}_{2}^{2}, \quad[3]=\mathfrak{p}_{3} \overline{\mathfrak{p}}_{3}, \quad \mathfrak{p}_{2} \mathfrak{p}_{3}=[\varepsilon+1], \quad \mathfrak{p}_{2} \overline{\mathfrak{p}}_{3}=[\varepsilon-1] .$$

  Deduce that $K$ has class number 2 .

  (c) Show that $\varepsilon$ is the fundamental unit of $K$. Hence prove that all solutions in integers $x, y$ of the equation $x^{2}-10 y^{2}=6$ are given by

  $$x+\sqrt{10} y=\pm \varepsilon^{n}\left(\varepsilon+(-1)^{n}\right), \quad n=0,1,2, \ldots$$

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• 4.II.20H

  Number Fields

  Part II, 2007

  Let $K$ be a finite extension of $\mathbb{Q}$ and let $\mathcal{O}=\mathcal{O}_{K}$ be its ring of integers. We will assume that $\mathcal{O}=\mathbb{Z}[\theta]$ for some $\theta \in \mathcal{O}$. The minimal polynomial of $\theta$ will be denoted by $g$. For a prime number $p$ let

  $$\bar{g}(X)=\bar{g}_{1}(X)^{e_{1}} \cdot \ldots \cdot \bar{g}_{r}(X)^{e_{r}}$$

  be the decomposition of $\bar{g}(X)=g(X)+p \mathbb{Z}[X] \in(\mathbb{Z} / p \mathbb{Z})[X]$ into distinct irreducible monic factors $\bar{g}_{i}(X) \in(\mathbb{Z} / p \mathbb{Z})[X]$. Let $g_{i}(X) \in \mathbb{Z}[X]$ be a polynomial whose reduction modulo $p$ is $\bar{g}_{i}(X)$. Show that

  $$\mathfrak{p}_{i}=\left[p, g_{i}(\theta)\right], \quad i=1, \ldots, r,$$

  are the prime ideals of $\mathcal{O}$ containing $p$, that these are pairwise different, and

  $$[p]=\mathfrak{p}_{1}^{e_{1}} \cdot \ldots \cdot \mathfrak{p}_{r}^{e_{r}}$$

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• 1.II.20G

  Number Fields

  Part II, 2008

  (a) Define the ideal class group of an algebraic number field $K$. State a result involving the discriminant of $K$ that implies that the ideal class group is finite.

  (b) Put $K=\mathbb{Q}(\omega)$, where $\omega=\frac{1}{2}(1+\sqrt{-23})$, and let $\mathcal{O}_{K}$ be the ring of integers of $K$. Show that $\mathcal{O}_{K}=\mathbb{Z}+\mathbb{Z} \omega$. Factorise the ideals [2] and [3] in $\mathcal{O}_{K}$ into prime ideals. Verify that the equation of ideals

  $$[2, \omega][3, \omega]=[\omega]$$

  holds. Hence prove that $K$ has class number 3 .

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• 2.II.20G

  Number Fields

  Part II, 2008

  (a) Factorise the ideals [2], [3] and [5] in the ring of integers $\mathcal{O}_{K}$ of the field $K=\mathbb{Q}(\sqrt{30})$. Using Minkowski's bound

  $$\frac{n !}{n^{n}}\left(\frac{4}{\pi}\right)^{s} \sqrt{\left|d_{K}\right|},$$

  determine the ideal class group of $K$.

  [Hint: it might be helpful to notice that $\frac{3}{2}=N_{K / \mathbb{Q}}(\alpha)$ for some $\left.\alpha \in K .\right]$

  (b) Find the fundamental unit of $K$ and determine all solutions of the equations $x^{2}-30 y^{2}=\pm 5$ in integers $x, y \in \mathbb{Z}$. Prove that there are in fact no solutions of $x^{2}-30 y^{2}=5$ in integers $x, y \in \mathbb{Z}$.

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• 4.II.20G

  Number Fields

  Part II, 2008

  (a) Explain what is meant by an integral basis of an algebraic number field. Specify such a basis for the quadratic field $k=\mathbb{Q}(\sqrt{2})$.

  (b) Let $K=\mathbb{Q}(\alpha)$ with $\alpha=\sqrt[4]{2}$, a fourth root of 2 . Write an element $\theta$ of $K$ as

  $$\theta=a+b \alpha+c \alpha^{2}+d \alpha^{3}$$

  with $a, b, c, d \in \mathbb{Q}$. By computing the relative traces $T_{K / k}(\theta)$ and $T_{K / k}(\alpha \theta)$, show that if $\theta$ is an algebraic integer of $K$, then $2 a, 2 b, 2 c$ and $4 d$ are rational integers. By further computing the relative norm $N_{K / k}(\theta)$, show that

  $$a^{2}+2 c^{2}-4 b d \text { and } 2 a c-b^{2}-2 d^{2}$$

  are rational integers. Deduce that $1, \alpha, \alpha^{2}, \alpha^{3}$ is an integral basis of $K$.

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• Paper 2, Section II, H

  Number Fields

  Part II, 2009

  Suppose that $K$ is a number field of degree $n=r+2 s$, where $K$ has exactly real embeddings.

  (i) Taking for granted the fact that there is a constant $C_{K}$ such that every integral ideal $I$ of $\mathcal{O}_{K}$ has a non-zero element $x$ such that $|N(x)| \leqslant C_{K} N(I)$, deduce that the class group of $K$ is finite.

  (ii) Compute the class group of $\mathbb{Q}(\sqrt{-21})$, given that you can take

  $$C_{K}=\left(\frac{4}{\pi}\right)^{s} \frac{n !}{n^{n}}\left|D_{K}\right|^{1 / 2}$$

  where $D_{K}$ is the discriminant of $K$.

  (iii) Find all integer solutions of $y^{2}=x^{3}-21$.

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• Paper 4, Section II, H

  Number Fields

  Part II, 2009

  Suppose that $K$ is a number field of degree $n=r+2 s$, where $K$ has exactly real embeddings.

  Show that the group of units in $\mathcal{O}_{K}$ is a finitely generated abelian group of rank at most $r+s-1$. Identify the torsion subgroup in terms of roots of unity.

  [General results about discrete subgroups of a Euclidean real vector space may be used without proof, provided that they are stated clearly.]

  Find all the roots of unity in $\mathbb{Q}(\sqrt{11})$.

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• Paper 1, Section II, H

  Number Fields

  Part II, 2009

  Suppose that $K$ is a number field with ring of integers $\mathcal{O}_{K}$.

  (i) Suppose that $M$ is a sub- $\mathbb{Z}$-module of $\mathcal{O}_{K}$ of finite index $r$ and that $M$ is closed under multiplication. Define the discriminant of $M$ and of $\mathcal{O}_{K}$, and show that $\operatorname{disc}(M)=r^{2} \operatorname{disc}\left(\mathcal{O}_{K}\right) .$

  (ii) Describe $\mathcal{O}_{K}$ when $K=\mathbb{Q}[X] /\left(X^{3}+2 X+1\right)$.

  [You may assume that the polynomial $X^{3}+p X+q$ has discriminant $-4 p^{3}-27 q^{2}$.]

  (iii) Suppose that $f, g \in \mathbb{Z}[X]$ are monic quadratic polynomials with equal discriminant $d$, and that $d \notin\{0,1\}$ is square-free. Show that $\mathbb{Z}[X] /(f)$ is isomorphic to $\mathbb{Z}[X] /(g)$.

  [Hint: Classify quadratic fields in terms of discriminants.]

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• Paper 1, Section II, G

  Number Fields

  Part II, 2010

  Suppose that $m$ is a square-free positive integer, $m \geqslant 5, m \not \equiv 1 \quad(\bmod 4)$. Show that, if the class number of $K=\mathbb{Q}(\sqrt{-m})$ is prime to 3 , then $x^{3}=y^{2}+m$ has at most two solutions in integers. Assume the $m$ is even.

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• Paper 2, Section II, G

  Number Fields

  Part II, 2010

  Calculate the class group of the field $\mathbb{Q}(\sqrt{-14})$.

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• Paper 4, Section II, G

  Number Fields

  Part II, 2010

  Suppose that $\alpha$ is a zero of $x^{3}-x+3$ and that $K=\mathbb{Q}(\alpha)$. Show that $[K: \mathbb{Q}]=3$. Show that $O_{K}$, the ring of integers in $K$, is $O_{K}=\mathbb{Z}[\alpha]$.

  [You may quote any general theorem that you wish, provided that you state it clearly. Note that the discriminant of $x^{3}+p x+q$ is $-4 p^{3}-27 q^{2}$.]

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• Paper 1, Section II, F

  Number Fields

  Part II, 2011

  Calculate the class group for the field $K=\mathbb{Q}(\sqrt{-17})$.

  [You may use any general theorem, provided that you state it accurately.]

  Find all solutions in $\mathbb{Z}$ of the equation $y^{2}=x^{5}-17$.

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• Paper 2, Section II, F

  Number Fields

  Part II, 2011

  (i) Suppose that $d>1$ is a square-free integer. Describe, with justification, the ring of integers in the field $K=\mathbb{Q}(\sqrt{d})$.

  (ii) Show that $\mathbb{Q}\left(2^{1 / 3}\right)=\mathbb{Q}\left(4^{1 / 3}\right)$ and that $\mathbb{Z}\left[4^{1 / 3}\right]$ is not the ring of integers in this field.

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• Paper 4, Section II, F

  Number Fields

  Part II, 2011

  (i) Prove that the ring of integers $\mathcal{O}_{K}$ in a real quadratic field $K$ contains a non-trivial unit. Any general results about lattices and convex bodies may be assumed.

  (ii) State the general version of Dirichlet's unit theorem.

  (iii) Show that for $K=\mathbb{Q}(\sqrt{7}), 8+3 \sqrt{7}$ is a fundamental unit in $\mathcal{O}_{K}$.

  [You may not use results about continued fractions unless you prove them.]

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• Paper 4, Section II, F

  Number Fields

  Part II, 2012

  Let $K=\mathbb{Q}(\sqrt{p}, \sqrt{q})$ where $p$ and $q$ are distinct primes with $p \equiv q \equiv 3(\bmod 4)$. By computing the relative traces $\operatorname{Tr}_{K / k}(\theta)$ where $k$ runs through the three quadratic subfields of $K$, show that the algebraic integers $\theta$ in $K$ have the form

  $$\theta=\frac{1}{2}(a+b \sqrt{p})+\frac{1}{2}(c+d \sqrt{p}) \sqrt{q}$$

  where $a, b, c, d$ are rational integers. Show further that if $c$ and $d$ are both even then $a$ and $b$ are both even. Hence prove that an integral basis for $K$ is

  $$1, \sqrt{p}, \frac{1+\sqrt{p q}}{2}, \frac{\sqrt{p}+\sqrt{q}}{2} .$$

  Calculate the discriminant of $K$.

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• Paper 2, Section II, F

  Number Fields

  Part II, 2012

  Let $K=\mathbb{Q}(\alpha)$ where $\alpha$ is a root of $X^{2}-X+12=0$. Factor the elements 2,3 , $\alpha$ and $\alpha+2$ as products of prime ideals in $\mathcal{O}_{K}$. Hence compute the class group of $K$.

  Show that the equation $y^{2}+y=3\left(x^{5}-4\right)$ has no integer solutions.

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• Paper 1, Section II, F

  Number Fields

  Part II, 2012

  Let $K$ be a number field, and $\mathcal{O}_{K}$ its ring of integers. Write down a characterisation of the units in $\mathcal{O}_{K}$ in terms of the norm. Without assuming Dirichlet's units theorem, prove that for $K$ a quadratic field the quotient of the unit group by $\{\pm 1\}$ is cyclic (i.e. generated by one element). Find a generator in the cases $K=\mathbb{Q}(\sqrt{-3})$ and $K=\mathbb{Q}(\sqrt{11})$.

  Determine all integer solutions of the equation $x^{2}-11 y^{2}=n$ for $n=-1,5,14$.

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• Paper 4, Section II, H

  Number Fields

  Part II, 2013

  State Dedekind's criterion. Use it to factor the primes up to 5 in the ring of integers $\mathcal{O}_{K}$ of $K=\mathbb{Q}(\sqrt{65})$. Show that every ideal in $\mathcal{O}_{K}$ of norm 10 is principal, and compute the class group of $K$.

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• Paper 2, Section II, H

  Number Fields

  Part II, 2013

  (i) State Dirichlet's unit theorem.

  (ii) Let $K$ be a number field. Show that if every conjugate of $\alpha \in \mathcal{O}_{K}$ has absolute value at most 1 then $\alpha$ is either zero or a root of unity.

  (iii) Let $k=\mathbb{Q}(\sqrt{3})$ and $K=\mathbb{Q}(\zeta)$ where $\zeta=e^{i \pi / 6}=(i+\sqrt{3}) / 2$. Compute $N_{K / k}(1+\zeta)$. Show that

  $$\mathcal{O}_{K}^{*}=\left\{(1+\zeta)^{m} u: 0 \leqslant m \leqslant 11, u \in \mathcal{O}_{k}^{*}\right\}$$

  Hence or otherwise find fundamental units for $k$ and $K$.

  [You may assume that the only roots of unity in $K$ are powers of $\zeta .$ ]

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• Paper 1, Section II, H

  Number Fields

  Part II, 2013

  Let $f \in \mathbb{Z}[X]$ be a monic irreducible polynomial of degree $n$. Let $K=\mathbb{Q}(\alpha)$, where $\alpha$ is a root of $f$.

  (i) Show that if $\operatorname{disc}(f)$ is square-free then $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.

  (ii) In the case $f(X)=X^{3}-3 X-25$ find the minimal polynomial of $\beta=3 /(1-\alpha)$ and hence compute the discriminant of $K$. What is the index of $\mathbb{Z}[\alpha]$ in $\mathcal{O}_{K}$ ?

  [Recall that the discriminant of $X^{3}+p X+q$ is $-4 p^{3}-27 q^{2}$.]

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• Paper 4, Section II, F

  Number Fields

  Part II, 2014

  Explain what is meant by an integral basis for a number field. Splitting into the cases $d \equiv 1(\bmod 4)$ and $d \equiv 2,3(\bmod 4)$, find an integral basis for $K=\mathbb{Q}(\sqrt{d})$ where $d \neq 0,1$ is a square-free integer. Justify your answer.

  Find the fundamental unit in $\mathbb{Q}(\sqrt{13})$. Determine all integer solutions to the equation $x^{2}+x y-3 y^{2}=17$.

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• Paper 2, Section II, F

  Number Fields

  Part II, 2014

  (i) Show that each prime ideal in a number field $K$ divides a unique rational prime $p$. Define the ramification index and residue class degree of such an ideal. State and prove a formula relating these numbers, for all prime ideals dividing a given rational prime $p$, to the degree of $K$ over $\mathbb{Q}$.

  (ii) Show that if $\zeta_{n}$ is a primitive $n$th root of unity then $\prod_{j=1}^{n-1}\left(1-\zeta_{n}^{j}\right)=n$. Deduce that if $n=p q$, where $p$ and $q$ are distinct primes, then $1-\zeta_{n}$ is a unit in $\mathbb{Z}\left[\zeta_{n}\right]$.

  (iii) Show that if $K=\mathbb{Q}\left(\zeta_{p}\right)$ where $p$ is prime, then any prime ideal of $K$ dividing $p$ has ramification index at least $p-1$. Deduce that $[K: \mathbb{Q}]=p-1$.

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• Paper 1, Section II, F

  Number Fields

  Part II, 2014

  State a result involving the discriminant of a number field that implies that the class group is finite.

  Use Dedekind's theorem to factor $2,3,5$ and 7 into prime ideals in $K=\mathbb{Q}(\sqrt{-34})$. By factoring $1+\sqrt{-34}$ and $4+\sqrt{-34}$, or otherwise, prove that the class group of $K$ is cyclic, and determine its order.

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• Paper 4, Section II, H

  Number Fields

  Part II, 2015

  Let $K$ be a number field. State Dirichlet's unit theorem, defining all the terms you use, and what it implies for a quadratic field $\mathbb{Q}(\sqrt{d})$, where $d \neq 0,1$ is a square-free integer.

  Find a fundamental unit of $\mathbb{Q}(\sqrt{26})$.

  Find all integral solutions $(x, y)$ of the equation $x^{2}-26 y^{2}=\pm 10$.

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• Paper 2, Section II, H

  Number Fields

  Part II, 2015

  (i) Let $d \equiv 2$ or $3 \bmod 4$. Show that $(p)$ remains prime in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ if and only if $x^{2}-d$ is irreducible $\bmod p$.

  (ii) Factorise $(2)$, (3) in $\mathcal{O}_{K}$, when $K=\mathbb{Q}(\sqrt{-14})$. Compute the class group of $K$.

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• Paper 1, Section II, $16 \mathrm{H}$

  Number Fields

  Part II, 2015

  (a) Let $K$ be a number field, and $f$ a monic polynomial whose coefficients are in $\mathcal{O}_{K}$. Let $M$ be a field containing $K$ and $\alpha \in M$. Show that if $f(\alpha)=0$, then $\alpha$ is an algebraic integer.

  Hence conclude that if $h \in K[x]$ is monic, with $h^{n} \in \mathcal{O}_{K}[x]$, then $h \in \mathcal{O}_{K}[x]$.

  (b) Compute an integral basis for $\mathcal{O}_{\mathbb{Q}(\alpha)}$ when the minimum polynomial of $\alpha$ is $x^{3}-x-4$.

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• Paper 2, Section II, F

  Number Fields

  Part II, 2016

  (a) Prove that $5+2 \sqrt{6}$ is a fundamental unit in $\mathbb{Q}(\sqrt{6})$. [You may not assume the continued fraction algorithm.]

  (b) Determine the ideal class group of $\mathbb{Q}(\sqrt{-55})$.

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• Paper 1, Section II, F

  Number Fields

  Part II, 2016

  (a) Let $f(X) \in \mathbb{Q}[X]$ be an irreducible polynomial of degree $n, \theta \in \mathbb{C}$ a root of $f$, and $K=\mathbb{Q}(\theta)$. Show that $\operatorname{disc}(f)=\pm N_{K / \mathbb{Q}}\left(f^{\prime}(\theta)\right)$.

  (b) Now suppose $f(X)=X^{n}+a X+b$. Write down the matrix representing multiplication by $f^{\prime}(\theta)$ with respect to the basis $1, \theta, \ldots, \theta^{n-1}$ for $K$. Hence show that

  $$\operatorname{disc}(f)=\pm\left((1-n)^{n-1} a^{n}+n^{n} b^{n-1}\right)$$

  (c) Suppose $f(X)=X^{4}+X+1$. Determine $\mathcal{O}_{K}$. [You may quote any standard result, as long as you state it clearly.]

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• Paper 4, Section II, F

  Number Fields

  Part II, 2016

  Let $K$ be a number field, and $p$ a prime in $\mathbb{Z}$. Explain what it means for $p$ to be inert, to split completely, and to be ramified in $K$.

  (a) Show that if $[K: \mathbb{Q}]>2$ and $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$ for some $\alpha \in K$, then 2 does not split completely in $K$.

  (b) Let $K=\mathbb{Q}(\sqrt{d})$, with $d \neq 0,1$ and $d$ square-free. Determine, in terms of $d$, whether $p=2$ splits completely, is inert, or ramifies in $K$. Hence show that the primes which ramify in $K$ are exactly those which divide $D_{K}$.

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• Paper 2, Section II, 18H

  Number Fields

  Part II, 2017

  (a) Let $L$ be a number field, $\mathcal{O}_{L}$ the ring of integers in $L, \mathcal{O}_{L}^{*}$ the units in $\mathcal{O}_{L}, r$ the number of real embeddings of $L$, and $s$ the number of pairs of complex embeddings of $L$.

  Define a group homomorphism $\mathcal{O}_{L}^{*} \rightarrow \mathbb{R}^{r+s-1}$ with finite kernel, and prove that the image is a discrete subgroup of $\mathbb{R}^{r+s-1}$.

  (b) Let $K=\mathbb{Q}(\sqrt{d})$ where $d>1$ is a square-free integer. What is the structure of the group of units of $K$ ? Show that if $d$ is divisible by a prime $p \equiv 3(\bmod 4)$ then every unit of $K$ has norm $+1$. Find an example of $K$ with a unit of norm $-1$.

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• Paper 1, Section II, H

  Number Fields

  Part II, 2017

  Let $\mathcal{O}_{L}$ be the ring of integers in a number field $L$, and let $\mathfrak{a} \leqslant \mathcal{O}_{L}$ be a non-zero ideal of $\mathcal{O}_{L}$.

  (a) Show that $\mathfrak{a} \cap \mathbb{Z} \neq\{0\}$.

  (b) Show that $\mathcal{O}_{L} / \mathfrak{a}$ is a finite abelian group.

  (c) Show that if $x \in L$ has $x \mathfrak{a} \subseteq \mathfrak{a}$, then $x \in \mathcal{O}_{L}$.

  (d) Suppose $[L: \mathbb{Q}]=2$, and $\mathfrak{a}=\langle b, \alpha\rangle$, with $b \in \mathbb{Z}$ and $\alpha \in \mathcal{O}_{L}$. Show that $\langle b, \alpha\rangle\langle b, \bar{\alpha}\rangle$ is principal.

  [You may assume that $\mathfrak{a}$ has an integral basis.]

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• Paper 4, Section II, H

  Number Fields

  Part II, 2017

  (a) Write down $\mathcal{O}_{K}$, when $K=\mathbb{Q}(\sqrt{\delta})$, and $\delta \equiv 2$ or $3(\bmod 4)$. [You need not prove your answer.]

  Let $L=\mathbb{Q}(\sqrt{2}, \sqrt{\delta})$, where $\delta \equiv 3(\bmod 4)$ is a square-free integer. Find an integral basis of $\mathcal{O}_{L} \cdot$ [Hint: Begin by considering the relative traces $t r_{L / K}$, for $K$ a quadratic subfield of $L .]$

  (b) Compute the ideal class group of $\mathbb{Q}(\sqrt{-14})$.

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• Paper 2, Section II, 20G

  Number Fields

  Part II, 2018

  Let $p \equiv 1 \bmod 4$ be a prime, and let $\omega=e^{2 \pi i / p}$. Let $L=\mathbb{Q}(\omega)$.

  (a) Show that $[L: \mathbb{Q}]=p-1$.

  (b) Calculate $\operatorname{disc}\left(1, \omega, \omega^{2}, \ldots, \omega^{p-2}\right)$. Deduce that $\sqrt{p} \in L$.

  (c) Now suppose $p=5$. Prove that $\mathcal{O}_{L}^{\times}=\left\{\pm \omega^{a}\left(\frac{1}{2}+\frac{\sqrt{5}}{2}\right)^{b} \mid a, b \in \mathbb{Z}\right\}$. [You may use any general result without proof, provided that you state it precisely.]

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• Paper 4, Section II, G

  Number Fields

  Part II, 2018

  Let $m \geqslant 2$ be a square-free integer, and let $n \geqslant 2$ be an integer. Let $L=\mathbb{Q}(\sqrt[n]{m})$.

  (a) By considering the factorisation of $(m)$ into prime ideals, show that $[L: \mathbb{Q}]=n$.

  (b) Let $T: L \times L \rightarrow \mathbb{Q}$ be the bilinear form defined by $T(x, y)=\operatorname{tr}_{L / \mathbb{Q}}(x y)$. Let $\beta_{i}=\sqrt[n]{m} i, i=0, \ldots, n-1$. Calculate the dual basis $\beta_{0}^{*}, \ldots, \beta_{n-1}^{*}$ of $L$ with respect to $T$, and deduce that $\mathcal{O}_{L} \subset \frac{1}{n m} \mathbb{Z}[\sqrt[n]{m}]$.

  (c) Show that if $p$ is a prime and $n=m=p$, then $\mathcal{O}_{L}=\mathbb{Z}[\sqrt[p]{p}]$.

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• Paper 1, Section II, G

  Number Fields

  Part II, 2018

  (a) Let $m \geqslant 2$ be an integer such that $p=4 m-1$ is prime. Suppose that the ideal class group of $L=\mathbb{Q}(\sqrt{-p})$ is trivial. Show that if $n \geqslant 0$ is an integer and $n^{2}+n+m<m^{2}$, then $n^{2}+n+m$ is prime.

  (b) Show that the ideal class group of $\mathbb{Q}(\sqrt{-163})$ is trivial.

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• Paper 4, Section II, 20G

  Number Fields

  Part II, 2019

  (a) Let $L$ be a number field, and suppose there exists $\alpha \in \mathcal{O}_{L}$ such that $\mathcal{O}_{L}=\mathbb{Z}[\alpha]$. Let $f(X) \in \mathbb{Z}[X]$ denote the minimal polynomial of $\alpha$, and let $p$ be a prime. Let $\bar{f}(X) \in(\mathbb{Z} / p \mathbb{Z})[X]$ denote the reduction modulo $p$ of $f(X)$, and let

  $$\bar{f}(X)=\bar{g}_{1}(X)^{e_{1}} \cdots \bar{g}_{r}(X)^{e_{r}}$$

  denote the factorisation of $\bar{f}(X)$ in $(\mathbb{Z} / p \mathbb{Z})[X]$ as a product of powers of distinct monic irreducible polynomials $\bar{g}_{1}(X), \ldots, \bar{g}_{r}(X)$, where $e_{1}, \ldots, e_{r}$ are all positive integers.

  For each $i=1, \ldots, r$, let $g_{i}(X) \in \mathbb{Z}[X]$ be any polynomial with reduction modulo $p$ equal to $\bar{g}_{i}(X)$, and let $P_{i}=\left(p, g_{i}(\alpha)\right) \subset \mathcal{O}_{L}$. Show that $P_{1}, \ldots, P_{r}$ are distinct, non-zero prime ideals of $\mathcal{O}_{L}$, and that there is a factorisation

  $$p \mathcal{O}_{L}=P_{1}^{e_{1}} \cdots P_{r}^{e_{r}}$$

  and that $N\left(P_{i}\right)=p^{\operatorname{deg} \bar{g}_{i}(X)}$.

  (b) Let $K$ be a number field of degree $n=[K: \mathbb{Q}]$, and let $p$ be a prime. Suppose that there is a factorisation

  $$p \mathcal{O}_{K}=Q_{1} \cdots Q_{s}$$

  where $Q_{1}, \ldots, Q_{s}$ are distinct, non-zero prime ideals of $\mathcal{O}_{K}$ with $N\left(Q_{i}\right)=p$ for each $i=$ $1, \ldots, s$. Use the result of part (a) to show that if $n>p$ then there is no $\alpha \in \mathcal{O}_{K}$ such that $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.

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• Paper 2, Section II, G

  Number Fields

  Part II, 2019

  (a) Let $L$ be a number field. State Minkowski's upper bound for the norm of a representative for a given class of the ideal class group $\mathrm{Cl}\left(\mathcal{O}_{L}\right)$.

  (b) Now let $K=\mathbb{Q}(\sqrt{-47})$ and $\omega=\frac{1}{2}(1+\sqrt{-47})$. Using Dedekind's criterion, or otherwise, factorise the ideals $(\omega)$ and $(2+\omega)$ as products of non-zero prime ideals of $\mathcal{O}_{K}$.

  (c) Show that $\mathrm{Cl}\left(\mathcal{O}_{K}\right)$ is cyclic, and determine its order.

  [You may assume that $\left.\mathcal{O}_{K}=\mathbb{Z}[\omega] .\right]$

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• Paper 1, Section II, 20G

  Number Fields

  Part II, 2019

  Let $K=\mathbb{Q}(\sqrt{2})$.

  (a) Write down the ring of integers $\mathcal{O}_{K}$.

  (b) State Dirichlet's unit theorem, and use it to determine all elements of the group of units $\mathcal{O}_{K}^{\times}$.

  (c) Let $P \subset \mathcal{O}_{K}$ denote the ideal generated by $3+\sqrt{2}$. Show that the group

  $$G=\left\{\alpha \in \mathcal{O}_{K}^{\times} \mid \alpha \equiv 1 \bmod P\right\}$$

  is cyclic, and find a generator.

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• Paper 2, Section II, 20G

  Number Fields

  Part II, 2020

  (a) Let $K$ be a number field of degree $n$. Define the discriminant $\operatorname{disc}\left(\alpha_{1}, \ldots, \alpha_{n}\right)$ of an $n$-tuple of elements $\alpha_{i}$ of $K$, and show that it is nonzero if and only if $\alpha_{1}, \ldots, \alpha_{n}$ is a $\mathbb{Q}$-basis for $K$.

  (b) Let $K=\mathbb{Q}(\alpha)$ where $\alpha$ has minimal polynomial

  $$T^{n}+\sum_{j=0}^{n-1} a_{j} T^{j}, \quad a_{j} \in \mathbb{Z}$$

  and assume that $p$ is a prime such that, for every $j, a_{j} \equiv 0(\bmod p)$, but $a_{0} \not \equiv 0\left(\bmod p^{2}\right)$.

  (i) Show that $P=(p, \alpha)$ is a prime ideal, that $P^{n}=(p)$ and that $\alpha \notin P^{2}$. [Do not assume that $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.]

  (ii) Show that the index of $\mathbb{Z}[\alpha]$ in $\mathcal{O}_{K}$ is prime to $p$.

  (iii) If $K=\mathbb{Q}(\alpha)$ with $\alpha^{3}+3 \alpha+3=0$, show that $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$. [You may assume without proof that the discriminant of $T^{3}+a T+b$ is $-4 a^{3}-27 b^{2}$.]

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• Paper 4, Section II, 20G

  Number Fields

  Part II, 2020

  Let $K$ be a number field of degree $n$, and let $\left\{\sigma_{i}: K \hookrightarrow \mathbb{C}\right\}$ be the set of complex embeddings of $K$. Show that if $\alpha \in \mathcal{O}_{K}$ satisfies $\left|\sigma_{i}(\alpha)\right|=1$ for every $i$, then $\alpha$ is a root of unity. Prove also that there exists $c>1$ such that if $\alpha \in \mathcal{O}_{K}$ and $\left|\sigma_{i}(\alpha)\right|<c$ for all $i$, then $\alpha$ is a root of unity.

  State Dirichlet's Unit theorem.

  Let $K \subset \mathbb{R}$ be a real quadratic field. Assuming Dirichlet's Unit theorem, show that the set of units of $K$ which are greater than 1 has a smallest element $\epsilon$, and that the group of units of $K$ is then $\left\{\pm \epsilon^{n} \mid n \in \mathbb{Z}\right\}$. Determine $\epsilon$ for $\mathbb{Q}(\sqrt{11})$, justifying your result. [If you use the continued fraction algorithm, you must prove it in full.]

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• Paper 1, Section II, 20G

  Number Fields

  Part II, 2020

  State Minkowski's theorem.

  Let $K=\mathbb{Q}(\sqrt{-d})$, where $d$ is a square-free positive integer, not congruent to 3 $(\bmod 4) .$ Show that every nonzero ideal $I \subset \mathcal{O}_{K}$ contains an element $\alpha$ with

  $$0<\left|N_{K / \mathbb{Q}}(\alpha)\right| \leqslant \frac{4 \sqrt{d}}{\pi} N(I)$$

  Deduce the finiteness of the class group of $K$.

  Compute the class group of $\mathbb{Q}(\sqrt{-22})$. Hence show that the equation $y^{3}=x^{2}+22$ has no integer solutions.

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• Paper 1, Section II, 20G

  Number Fields

  Part II, 2021

  Let $K=\mathbb{Q}(\alpha)$, where $\alpha^{3}=5 \alpha-8$

  (a) Show that $[K: \mathbb{Q}]=3$.

  (b) Let $\beta=\left(\alpha+\alpha^{2}\right) / 2$. By considering the matrix of $\beta$ acting on $K$ by multiplication, or otherwise, show that $\beta$ is an algebraic integer, and that $(1, \alpha, \beta)$ is a $\mathbb{Z}$-basis for $\mathcal{O}_{K} \cdot$ [The discriminant of $T^{3}-5 T+8$ is $-4 \cdot 307$, and 307 is prime.]

  (c) Compute the prime factorisation of the ideal (3) in $\mathcal{O}_{K}$. Is (2) a prime ideal of $\mathcal{O}_{K} ?$ Justify your answer.

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• Paper 2, Section II, 20G

  Number Fields

  Part II, 2021

  Let $K$ be a field containing $\mathbb{Q}$. What does it mean to say that an element of $K$ is algebraic? Show that if $\alpha \in K$ is algebraic and non-zero, then there exists $\beta \in \mathbb{Z}[\alpha]$ such that $\alpha \beta$ is a non-zero (rational) integer.

  Now let $K$ be a number field, with ring of integers $\mathcal{O}_{K}$. Let $R$ be a subring of $\mathcal{O}_{K}$ whose field of fractions equals $K$. Show that every element of $K$ can be written as $r / m$, where $r \in R$ and $m$ is a positive integer.

  Prove that $R$ is a free abelian group of $\operatorname{rank}[K: \mathbb{Q}]$, and that $R$ has finite index in $\mathcal{O}_{K}$. Show also that for every nonzero ideal $I$ of $R$, the index $(R: I)$ of $I$ in $R$ is finite, and that for some positive integer $m, m \mathcal{O}_{K}$ is an ideal of $R$.

  Suppose that for every pair of non-zero ideals $I, J \subset R$, we have

  $$(R: I J)=(R: I)(R: J) .$$

  Show that $R=\mathcal{O}_{K}$.

  [You may assume without proof that $\mathcal{O}_{K}$ is a free abelian group of rank $[K: \mathbb{Q}]$ ] ]

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• Paper 4 , Section II, 20G

  Number Fields

  Part II, 2021

  (a) Compute the class group of $K=\mathbb{Q}(\sqrt{30})$. Find also the fundamental unit of $K$, stating clearly any general results you use.

  [The Minkowski bound for a real quadratic field is $\left|d_{K}\right|^{1 / 2} / 2 .$ ]

  (b) Let $K=\mathbb{Q}(\sqrt{d})$ be real quadratic, with embeddings $\sigma_{1}, \sigma_{2} \hookrightarrow \mathbb{R}$. An element $\alpha \in K$ is totally positive if $\sigma_{1}(\alpha)>0$ and $\sigma_{2}(\alpha)>0$. Show that the totally positive elements of $K$ form a subgroup of the multiplicative group $K^{*}$ of index 4 .

  Let $I, J \subset \mathcal{O}_{K}$ be non-zero ideals. We say that $I$ is narrowly equivalent to $J$ if there exists a totally positive element $\alpha$ of $K$ such that $I=\alpha J$. Show that this is an equivalence relation, and that the equivalence classes form a group under multiplication. Show also that the order of this group equals

  $$\begin{cases}\text { the class number } h_{K} \text { of } K & \text { if the fundamental unit of } K \text { has norm }-1 \\ 2 h_{K} & \text { otherwise. }\end{cases}$$

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