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September 12th, 2018, 01:17 PM  #1 
Newbie Joined: Jul 2018 From: morocco Posts: 21 Thanks: 0 Math Focus: algebraic number theory  Help please. Hilbert symbol
HELLO 1) Let K be a number field, p a rational prime and P a ramified prime ideal of K laying over p. Is it true that: (x,p)_P=(x/P) where (.,.)_P is the quadratic Hilbert symbol and (./P) is the second power residue symbol and x is an element of K (or a unit of K) ?? 2) Can you please guide me to a reference in which I can find such kind of properties of Hilbert symbol (except Gras)? Thank you very much. Last edited by skipjack; September 12th, 2018 at 11:06 PM. 
September 12th, 2018, 08:16 PM  #2 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 286 Thanks: 88 Math Focus: Number Theory, Algebraic Geometry 
This isn't true in general. For example, if $K = \mathbb{Q}(\sqrt{p})$ and $P = (\sqrt{p})$, then $p$ is a square in $K$ and so $(x,p)_P = 1$ always (while $(x/P)$ will often be $1$). Unfortunately, the texts I know on this topic seem either too basic or too advanced for your needs, but I'll mention a couple just in case they prove helpful. Serre's "A Course in Arithmetic" deals nicely with the case $K = \mathbb{Q}$. Your result is true here, and is proven as case 2) of the proof of theorem 1 in chapter 3. For a much more thorough account of the Hilbert symbol, Neukirch's "Class Field Theory" is a great reference. However, rather than giving an elementary formulation of the quadratic Hilbert symbol (in terms of solutions to a quadratic equation) as you are probably used to, it goes straight to defining the nth power Hilbert symbol using local class field theory. Here you'll in fact see that $(x,\pi)_P = (x/P)$ whenever $\pi \in K_P$ is a uniformizer and $x \in U_P$, where $K_P$ is a completion of $K$ at $P$ and $U_P$ is its group of units. (Actually, this is how the nth power residue symbol is defined in this book, but then it's immediately proven that this coincides with the usual definition.) 
September 13th, 2018, 03:11 AM  #3 
Newbie Joined: Jul 2018 From: morocco Posts: 21 Thanks: 0 Math Focus: algebraic number theory 
So much interesting ! thank you very much 

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