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Results tagged with algebraic-number-theory
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user 17907
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
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 …
- 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 …
- 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+ …
- 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 …
- 4,745
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 …
- 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 …
- 4,745
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 …
- 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^ …
- 4,745