# Questions tagged [hilbert-symbol]

The hilbert-symbol tag has no usage guidance.

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### Does $17x^4+y^2=-1$ have solution in $\Bbb{Q}_2(\sqrt{-5})$?

This question raised when I tried to calculate $2$-Selmer group of elliptic curve $E:y^2=x^3+17x$ over $\Bbb{Q}(\sqrt{-5})$.
$17x^4+y^2=-1$ does not have solution in $\Bbb{Q}_2$
(https://math....

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### Hilbert symbol over 2-adic field

Let $F$ be a finite extension of $\mathbb{Q}_2$, and let $(-,-)_F$ be the quadratic Hilbert symbol over $F$. Is the following true?
$(-1,-1)_F=1$ if and only if $\sqrt{-1}\in F$

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### Relation in Brauer group coming from trace form

Let $L/K$ be a cubic (or, more generally, odd-order) extension of fields of characteristic $0$. To every element $a \in L^\times$ we can associate the quadratic form
\begin{align*}
q_a : L &\to K \...

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### Hilbert Symbols, Norms, and p-adic roots of unity

Let $p$ be an odd prime number,
let $\mathbb{Q}_p$ be the field of $p$-adic numbers,
and let $\overline{\mathbb{Q}_p}$ be an algebraic closure of it.
For a primitive $p$-th root of unity $\zeta_p \in ...

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### Dyadic Hilbert symbols and higher unit groups

Let $F$ be a local dyadic number field, $\mathfrak{p}$ its maximal ideal, $(*,*)_F$ its quadratic Hilbert symbol and $e$ its ramification index (i.e. $\mathfrak{p}^e$ is exact divisor of $2$). Fix an ...

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### Hilbert symbol averages

Let me call a pair of integers $a, b$ acceptable if the equation $ax^2 + by^2 = z^2$ has a non-trivial rational solution. Theorem 4.5.4 of Cojocaru-Murty's book on Sieves says that the number of ...

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### Norm Residue Symbol refinement?

From Wikipedia: given $a\in K^\times$,
(a,b)=1 for all b [in K*] if and only if a is in K*ⁿ
So suppose that $(\frac{a\ ,\ K^\times\!}{p})\neq 1$ [assume $n$ above ...

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### Skew symmetry for the Hilbert symbol

Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ $$(a,...

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### Weil reciprocity vs Artin reciprocity

This is probably an easy question for the experts:
Given two rational functions $f$, $g$ on a non-singular projective algebraic curve X (over an algebraically closed field $k$) and $p \in X$, one ...

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### Hilbert symbol and Weil index, beyond the quadratic case?

Let $F$ be a local nonarchimedean field. Let $n$ be a positive integer for which the group $\mu_n(F)$ of $n^{th}$ roots of unity in $F$ has order $n$. Let $\epsilon: \mu_n(F) \rightarrow C^\times$ ...