# Tagged Questions

Questions related to the algebraic structure of algebraic integers

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### Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, K, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n$$ ...
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### confusion with calculating the ideal class group of a quadratic field

I am a bit confused with the procedure of calculating the ideal class group of a quadratic field. From what I understood the computation starts by finding the Minkowski's bound say $n$. Then we list ...
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### Find all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$.

I am studying Rational Points on Elliptic Curves by Silverman and Tate. This is Problem 2.12 (h). Determine all of the points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. Also ...
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### Rational prime with specified factorization in $\mathbf{Z}[\mu_q]$

Let $r$ and $f$ be given positive integers. Prove that there exist primes $p$ and $q$ such that $p\mathbf{Z}[\mu_q]$ (where $\mu_q$ is a primitive $q$th root of unity) is a product of exactly $r$ ...
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### When is $\mathbb{Z} [x]/f(x)$ a Dedekind domain?

Given a monic separable irreducible polynomial $f$ with integer coefficients, when $\mathbb{Z} [x]/f(x)$ is a Dedekind domain? And when it happens to be a Dedekind domain, how to know its class ...
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### Arithmetic in a dihedral extension

Let $\;$L = $\Bbb Q$[$\sqrt[4]{2}$, i ]$\;$ which is a dihedral extension of the rationals. There are three quadratic and five quartic intermediate fields between L and $\Bbb Q$. The following ...
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### ramification index in an example

Let $L=\mathbb{Q}_5[x]/(x^4+5x^2+5)$, where $\mathbb{Q}_5$ is the field of 5-adic numbers. Note that the polynomial that we are quotienting out by is an Eisenstein polynomial. So $L/\mathbb{Q}_5$ is a ...
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### Neukirch Abstract Kummer Theory. Understanding a Proof.

This question is a sort of follow up to this question, where I introduced context. Neukirch mysterious homomorphism in Abstract Kummer Theory (in his book ANT) The thing I don't understand know is ...
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### $p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
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### Minimal polynomial of root of unity over quadratic field

Let $p$ be an odd prime and consider the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$ and its quadratic subfield $\mathbb{Q}(\sqrt{\pm p})=:K$. I am interested in the minimal polynomial of a root of ...
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### $p = x^2 + xy + 3y^2$ if and only if $p \equiv 1$, $3$, $4$, $5$, $9$ mod $11$? [duplicate]
For a prime number $p \neq 11$, do we have $p = x^2 + xy + 3y^2$ for some $x$, $y \in \mathbb{Z}$ if and only if $p \equiv 1, 3, 4, 5, 9$ mod $11$? An example where this is true:5 = 1^2 + 1 \times ...
$\zeta_q\in L^{G({\frak P})}\Leftrightarrow$ if $\sigma\in$ $G(\frak p$) then $\sigma(\zeta_q)=\zeta_q\Leftrightarrow$ if $\sigma$ fixes $\frak P$ then $\sigma$ fixes $\zeta_q$. What's next? Suppose ...