Recitation talks for math 223a Algebraic Number Theory, Harvard, Fall 2014

Yihang Zhu

• 9/22/14. We proved the field of $p$-adic complex numbers $\mathbb C_p$ is algebraically closed.
• 9/29/14. We talked about the Theorem of Tate and Ax that determines the fixed subfield of $\mathbb C_p$ by a closed subgroup of $G_{\mathbb Q_p}$. Notes for the two talks.
• 10/20/14. We talked about Kummer extensions and Artin-Schreier extensions. We proved the additive and multiplicative versions of Satz 90 (aka Hilbert's Theorem 90).
• 11/2/14. We talked about group cohomology informally, and gave a precise definition of $H^1$ in terms of cocycles. We showed that the cohomological version of Satz 90 implies the classical version, by explicitly computing $H^1$ of a cyclic group. Then we proved cohomological Satz 90 using the "Poincare series", following Serre's book Local Fields.
• 11/10/14. We talked about the local norm residue symbol. Here is a brief summary:
  We first stated the main theorem of Kummer theory for general fields. For $m$ an integer invertible in a field $K$ containing all the $m$-th roots of unity, Kummer theory classifies all the abelian extensions of $K$ whose Galois groups are killed by $m$. Specialize to the case $K$=local field, we obtain a perfect pairing $K^*/{K^*}^m \times G_K^{ab}/m \to \mu_m,$ between two finite abelian groups. Composing with the local Artin map, we obtain a perfect pairing $K^*/{K^*}^m \times K^*/{K^*}^m \to \mu_m,$ called the norm-residue symbol. Next we used knowledge about the local artin map to deduce formal properties of the norm-residue symbol. When $m=2$, the norm-residue symbol is classically known as the Hilbert symbol, and $(a,b) =1$ if and only if $ax^2 + by^2 = z^2$ has a non-trivial solution in $K$. In general, when $m$ is not divisible by the residue characteristic, we deduced an explicit formula for the symbol, (called the tame symbol in this case.) Finally, we showed how to use the above theory to prove statements such as $5x^2 + 6 y^2 =1$ has no solution in the rationals. The main reference is Chapter 5.3 of Neukirch: Algebraic number theory. Note that Neukirch calls our symbol "the Hilbert symbol", and he uses the term "norm residue symbol" to mean the local artin map.
• 11/17/14 We talked about the general power reciprocity laws. This is based on the local discussion last time and Artin's reciprocity. In particular, concrete calculation of certain local norm residue symbols gives rise to classical reciprocity laws, such as quadratic and cubic reciprocity. These notes contain complete computation for cubic reciprocity omitted in the talk.