Questions tagged [algebraic-number-theory]
Questions related to the algebraic structure of algebraic integers
8,153 questions
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Does there exist a prime decomposition for every possible combination of ramification and inertia degree
I just finished the treatment of quadratic fields and cyclotomic fields in Marcus' Number Fields and I decided to approach biquadratic fields as a fun exercise. From the Ram-Rel identity we know that ...
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Class group of $\mathbb{Q}(\sqrt[3]{3})$
I am reading this post : Ideal Class Group of $ \mathbb{Q}(\sqrt[3]{3}) $ and I don't understand some one part. I hope this post isn't considered as a duplicate :) In book ' A conversational ...
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Quintic residues and $p=x^2+125y^2$
Euler conjectured (and Gauss later proved) that:
If $p\equiv 1\pmod 3$, $2$ is a cubic residue mod $p$ iff $p=x^2+27y^2$ for some integer $x$ and $y$.
If $p\equiv 1\pmod 4$, $2$ is a quartic residue ...
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Which permutations Galois considered out of $n!$ permutations in $S_n$ to prove insolvability of quintic polynomial?
It seems to me that the beginning of Galois theory (Galois's work) is still not clear to me. To prove the quintic or higher degree polynomial is not solvable in radicals, Evertise Galois considered ...
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Do places ever become inert in the Hermitian function field over $\mathbb{F}_4$ ($q = 2$)?
I consider the Hermitian function field
$H = \mathbb{F}_4(x,y)$, given by $y^2 + y = x^3$,
which is a quadratic extension of $F = \mathbb{F}_4(x)$.
Let $P$ be a place of $F$. If $v_P(x^3) \ge 0$, then ...
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Computing the Schirokauer map in Number Field Sieve for Discrete Logarithms
I'm following the Worked Example for the Special Number Field Sieve paper and I got stuck at the Schirokauer Map construction.
Could someone verify my map construction logic and help me answer these ...
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Existence of unramified cyclotomic extensions of number fields
Let $p$ be an odd prime and $\zeta_{p}$ be a primitive $p$-th root of unity. Is there a number field $K$ such that $\zeta_{p}\notin K$ and $K(\zeta_{p})/K$ is unramified at any prime of $K$?
I know ...
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Existence of rational elliptic curves with trivial torsion over $\mathbb{Q}$ acquiring 5-torsion over a cubic field
I am investigating the torsion growth of rational elliptic curves upon base change to cubic fields.
Specifically, I am looking for a rational elliptic curve $E/\mathbb{Q}$ with trivial rational ...
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Splitting field of a polynomial over $\mathbb{Q}_p$?
How to find the degree and the ramification index of the splitting field of a certain polynomial over $\mathbb{Q}_p$? For instance, $f(x)=3+9x+3x^4+x^6$ over $\mathbb{Q}_3$?
If $\gamma$ is one of the ...
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The $p$-adic valuations of the pairwise differences for all distinct pairs of roots of an irreducible polynomial
Let $f$ be an irreducible polynomial over $\mathbb Q_p$ and $\alpha_1,\ldots,\alpha_n$ be its roots is an algebraic closure. It is known that the valuations of $\alpha_i$ are all equal. Is it true ...
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Computing large division fields of elliptic curves with CM
Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by $\mathbb{Q}(i)$, e.g. any curve of the form $E:y^2 = x^3 + Ax$ with $A \in \mathbb{Q}^\times$.
I would like to compute generators ...
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Calculating an integral basis for ring of integers in quadratic fields through the discriminant of basis
It is known that for squarefree $m$, in $\mathbb Q(\sqrt{m})$ the ring of integers is $\{a+b\sqrt{m} : a,b \in \mathbb{Z}\}$ if $m \equiv 2,3 \bmod 4$ and is $\{\frac{a}{2}+\frac{b}{2}\sqrt{m} : a,b \...
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Number of factors of a polynomial through Newton polygon argument
Let the Newton polygon of $f(x) \in \mathbb{Q}_p[x]$ has a single segment of horizontal length $L$ and slope $\lambda=\frac{h}{e},~\gcd(h,e)=1$.
Is it true that $f(x)$ has $\frac{L}{e}$ many ...
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Complement direct summand to a non-principal ideal in a number ring
Let $K$ be a number field, that is, a finite extension of ${\mathbb Q}$. Let $R={\mathcal O}_K$ be the Dedekind domain of algebraic integers in $K$.
It is well-known that an ideal $I\subset R$ is a ...
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Two questions on Deuring's Lifting Theorem
I have two questions about Deuring's Lifting Theorem:
Let $E/\mathbb F_q$ be an elliptic curve over a finite field and let $\phi \in End(E)$ be nonzero. There exists an elliptic curve $E^*$ over a ...
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