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Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces

1 vote

Irreducibility of polynomials over some number fields

Here is a different approach, which is arguably a bit more elementary. If $f=X^n-p$ splits in $K$, and $g$ is one of its factors, then the constant term of $g$, being a product of zeros of $f$, must b …
R.P.'s user avatar
  • 4,745
7 votes
Accepted

Square root in number field

This answer is meant to answer only your second question. Claim. Let $K=\mathbb{Q}(\sqrt[3]{2})$ and $\alpha = \sqrt[3]{2}-\sqrt[3]{4} \in K$. Then there does not exist $\beta \in K$ such that $\beta^ …
R.P.'s user avatar
  • 4,745
6 votes
Accepted

Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$?

emtom has found the right reference, but there is a more explicit result in that book (Ireland and Rosen, A Classical Introduction to Modern Number Theory). In fact, Theorem 5 of Chapter 8 (on page 10 …
R.P.'s user avatar
  • 4,745
7 votes
Accepted

Which quartic fields contain the 4th roots of unity in their Galois closure?

As explained in the comments, the only non-trivial case is where $K/\mathbb{Q}$ has Galois closure $K'/\mathbb{Q}$ with $\operatorname{Gal}(K'/\mathbb{Q})$ isomorphic to $D_4$. I will do this case her …
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6 votes
Accepted

Reference request to proof that H$^2(\Gamma, \mathbb{Q}/\mathbb{Z}) = 0$

By the Galois cohomology long exact sequence, this is isomorphic to $\operatorname{H}^3(\Gamma,\mathbb{Z})$, and the vanishing of this is Chapter I, Corollary 4.17 in Milne's Arithmetic Duality Theore …
R.P.'s user avatar
  • 4,745
8 votes
Accepted

$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$

We need to distinguish according to whether $p$ is congruent to $1$ or $3$ modulo $4$, and whether $-b/a$ is or is not a fourth power modulo $p$. (Note that the case $p=2$ is trivial since $ax^4+by^4+ …
R.P.'s user avatar
  • 4,745
4 votes

Analogy between the nodal cubic curve $y^2=x^3+x^2$ and the ring $\mathbb{Z}[\sqrt{-3}]$?

By a result that I know by the name of the "Kummer-Dedekind theorem" (see Chapter 3 of these notes), there is a unique prime ideal $\mathfrak{p}=(2,1+\sqrt{3})$ of $R = \mathbb{Z}[\sqrt{-3}]$ lying ov …
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  • 4,745
12 votes

Is being principal a local property?

Only in principal ideal domains (PIDs). If by number ring you mean Dedekind domain, then all its localizations at prime ideals are discrete valuation rings (except the one at 0 which is a field), whic …
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  • 4,745