189-596A: Class Field Theory

Assignment 1. Due: Monday, September 24.

1. If $q$ is a prime that is congruent to $1$ modulo $4$, show that $\mathbb Q(\sqrt{q})$ is contained in the cyclotomic field $\mathbb Q(\zeta_q)$ by computing the square of the appropriate Gauss sum.

2. (a) Show that the field $\mathbb Q_p(\zeta_p)$ is a Kummer extension of $\mathbb Q_p$, by showing that $\mathbb Q_p$ contains a $(p-1)$-st root of unity.

(b) Show that there is an injective homomorphism $\chi: (\mathbb Z/p\mathbb Z)^\times \rightarrow \mathbb Q_p^\times$.

(c) Show ``by pure thought" that the element $$ \alpha := \sum_{a=1}^{p-1} \chi(a) \zeta_p^a \in \mathbb Q_p(\zeta_p)$$ satisfies $\alpha^{p-1}\in\mathbb Q_p$. Explain why $\alpha^{p-1}$ even belongs to $\mathbb Z[\zeta_{p-1}]$.

(d) Let $a$ be ~~the norm from~~ $\mathbb Q(\zeta_{p-1})$ to $\mathbb Q$ of $\alpha^{p-1}$. Show that $a$ belongs to $\mathbb Q_p$ and that $\mathbb Q_p(\zeta_p) = \mathbb Q_p( a^{1/(p-1)})$.
(Thanks to James for pointing out a silly typo in this question.)

(e) Compute the value of $a$, and show that $a/p^{p-2}$ is an element of $\mathbb Z_p$ that is congruent to $-1$ modulo $p$. Conclude that $\mathbb Q_p(\zeta_p) = \mathbb Q_p((-p)^{1/(p-1)})$, as asserted in Kedlaya's notes.
(Thanks to James for pointing out a more substantial error in an earlier version of this question: it should be $a/p^{p-2}$ and not $a/p$ as was previously stated.)

3. Let $K$ be a field of characteristic zero, let $L=K(\zeta_n)$, and let $G= {\rm Gal}(L/K)$. Prove the $G$-equivariance of the Kummer pairing $$ \langle \ , \ \rangle: {\rm Gal}(L^{\rm ab}/L) \times L^\times/(L^\times)^n \rightarrow \mu_n$$ that was asserted in class, i.e., that $$\langle ghg^{-1}, g \alpha\rangle = g \langle h, \alpha\rangle, \quad \mbox{ for all } g\in G, \quad h\in {\rm Gal}(L^{\rm ab}/L), \quad \alpha\in L^\times.$$

4. Let $K/K_0$ be a totally ramified extension of local fields of residue characteristic $p$. If $u\in \mathcal O_K^\times$ is a unit of the ring of integers of $K$, and $e$ is an integer that is prime to $p$, show that $u$ can be written as $$ u = u_0 v^e, \qquad \mbox{ where } u_0\in \mathcal O_{K_0}^\times, \quad v\in \mathcal O_K^\times. $$ Conclude that, as asserted in class, the field $K$ has a uniformizer $\pi_K$ for which $\pi_K^e$ is a uniformiser of $K_0$.

5. Problem (3) of Kedlaya, page 5.

6. Problem (4) of Kedlaya, page 5.

7. Problem (2) of Kedlaya, page 9.

8. Problem (2) of Kedlaya, page 13.

9. Problems (3) and (4) of Kedlaya, page 13.

10. Problems (5) and (6) of Kedlaya, page 13.