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September 12th, 2018, 01:17 PM   #1
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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.
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September 12th, 2018, 08:16 PM   #2
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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 n-th 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 n-th power residue symbol is defined in this book, but then it's immediately proven that this coincides with the usual definition.)
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September 13th, 2018, 03:11 AM   #3
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So much interesting !
thank you very much
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