# Algebraïsche getaltheorie

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# Examenvragen

## Exam January 2018

All questions were open book. Questions 1 and 2 had to explained orally.

### Question 1

• In the proof of Theorem 4.16, why is (O_K)/(pO_K) isomorphic to (Z/pZ)^n?
• In the proof of Lemma 5.13, why does the case I=O_K suffice?
• In the first lines section 8.1, why is Phi_m(X) in Z[X]?
• In the proof of Theorem 8.10, why is pa_i = Tr(\alpha \zeta_p^{-i} - \alpha \zeta)?

### Question 2

For this question, a formula for the discriminant of a degree 3 polynomial was given on the blackboard. Let f_1 = X^3+X+1, f_2 = X^3+X^2-2X+1. Let a_i be a root of f_i.

• Show that f_1 and f_2 are irreducible over Q.
• Let K_i be Q(a_i), show that O_{K_i} = Z[a_i] for i=1,2.
• Show that the groups of units of O_{K_1} and O_{K_2} are isomorphic.
• Give a method for determining whether K_1 and K_2 are isomorphic as fields.

### Question 3

Let f = prod(X-theta_i, i=1..n) be a monic polynomial over Q. Its discriminant is defined as disc(f) = prod((theta_i - theta_j)^2, 1\leq i<j\leq n). Let m be an integer and define f_m = X^3+(m+3)X^2+mX-1. We have disc(f_m)=(m^2+3m+9)^2. Let a_m be a root of f_m.

• Show that f_m is irreducible over Q and that K_m = Q(a_m) is the splitting field of f, for all m. (Hint: let f be an irreducible of degree n in Q[X] and let K be its splitting field. Then Gal(K/Q)\subseteq A_n iff disc(f) is a square in Q).
• Describe the group of units in O_{K_m}.
• Find sufficient conditions on m such that O_{K_m} = Z[a_m]. (Hint: how are disc(f_m) and Delta_{K_m/Q}(1, a_m, a_m^2) related?)

Now fix an integer s such that the condition from the last point holds, put K = K_s and f_s = f.

• Find the set of primes S that ramify in K.
• For p not in S, what are the possible factorizations of pO_K in O_K? Give an example of each of those. What can you say about the factorization of f modulo p?
• (Bonus) Prove the hint in the first point of this exercise.

### Question 4

Compute the class group of Q(sqrt(-30)) and give a representative of each in element in the group.

## Examen van 29 augustus 2008

De eerste vraag is gesloten boek.

• Veronderstel dat ${\displaystyle [K:\mathbb {Q} ]=n}$. Bewijs dat ${\displaystyle \alpha _{1},\alpha _{2},\,\cdots ,\alpha _{n}\in K}$ lineair onafhankelijk zijn over ${\displaystyle \mathbb {Q} }$ als en slechts als ${\displaystyle \Delta \left(\alpha _{1},\alpha _{2},\,\cdots ,\alpha _{n}\right)\neq 0}$.
• Ontbind ${\displaystyle 2{\mathcal {O}}_{\mathbb {Q} \left(\xi _{23}\right)}}$ in priemidealen in ${\displaystyle {\mathcal {O}}_{\mathbb {Q} \left(\xi _{23}\right)}}$. Hint: probeer met behulp van kwadratische Gauss-sommen in te zien dat ${\displaystyle \mathbb {Q} \left({\sqrt {-23}}\right)\subseteq \mathbb {Q} \left(\xi _{23}\right)}$.
• Zij ${\displaystyle n=p^{\alpha }}$ met ${\displaystyle p}$ priem. Bewijs dat er een priemideaal ${\displaystyle M}$ van graad 1 in ${\displaystyle {\mathcal {O}}_{\mathbb {Q} \left(\xi _{n}\right)}}$ bestaat zodat ${\displaystyle p{\mathcal {O}}_{\mathbb {Q} \left(\xi _{n}\right)}=M^{\phi (n)}}$ en vind ${\displaystyle M}$.
• Zij ${\displaystyle n=p^{\alpha }m}$ met ${\displaystyle p}$ priem en ${\displaystyle ggd(p,m)=1}$. Zij ${\displaystyle P}$ een priemideaal van ${\displaystyle {\mathcal {O}}_{\mathbb {Q} \left(\xi _{n}\right)}}$ boven ${\displaystyle p}$.
• Bepaal expliciet de ramificatie-index ${\displaystyle e_{P}}$, de graad ${\displaystyle f_{P}}$ en het aantal priemidealen ${\displaystyle g}$ boven ${\displaystyle p}$. Hint: vind ondergrenzen en bewijs gelijkheid.
• We weten dat ${\displaystyle \Gamma \left(\mathbb {Q} \left(\xi _{n}\right):\mathbb {Q} \right)\cong \mathbb {Z} _{n}^{\times },\cdot \cong \mathbb {Z} _{m}^{\times },\cdot \times \mathbb {Z} _{p^{\alpha }}^{\times },\cdot }$. Beschrijf de inertie- en decompositiegroepen van ${\displaystyle P}$ m.b.v. de gegeven multiplicatieve groepen.

## Examen van 8 juni 2009

1. Zij K een getallenveld. Toon aan dat er een integrale basis voor ${\displaystyle {\mathcal {O}}_{K}}$ bestaat. [enige vraag die gesloten boek was]
2. Wat is de structuur van de groep van de ideaalklassen van ${\displaystyle \mathbb {Z} [{\sqrt {-21}}]}$?
3. ...
4. Geef alle priemgetallen p waarboven juist 4 priemidealen liggen. [Over welk getallenveld gaat dit?]
5. ...