Questions tagged [algebraic-number-theory]
Questions related to the algebraic structure of algebraic integers
8,150 questions
0 votes
0 answers
31 views
Elementary Proof that the algebraic integers form a ring. [duplicate]
Proving the algebraic integers form a ring is very easy with a little bit of Module Theory, and the Cayley-Hamiltonian Theorem. However I am looking for an elementary proof. I have proven, in an ...
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2 votes
1 answer
67 views
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 ...
- 149
1 vote
1 answer
50 views
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 ...
- 1,259
1 vote
2 answers
64 views
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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2 votes
1 answer
80 views
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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1 vote
2 answers
82 views
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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2 votes
1 answer
97 views
Weak cubic reciprocity in Diamond and Shurman
In Diamond and Shurman, I am trying to understand the weak form of cubic reciprocity in section 4.11 - how it comes from the more standard version. In the following $A$ is the Eisenstein integers, and ...
0 votes
0 answers
88 views
Why is the derivative of minimal polynomial contained in different?
I am solving exercise 3.37 (e) in Number Fields by Marcus. Let $S/R$ be an extension of Dedekind domains, $P$ is a prime of $R$ and $Q$ is a prime of $S$ lying over $P$. Let $e=e(Q|P)$, then in (d) I ...
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