# Tagged Questions

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### Show that the HNF of the column vector $[a_{1},…,a_{n}]^{T}$ is exactly $[gcd(a_{1},…,a_{n}),0,…,0]^{T}$

HNF = Hermite Normal Form. I see why this is true by computing an example...and I know I need to use the Euclidean Algorithm...
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### Ideal in Dedekind domain

Let $D$ be Dedekind domain and $I$ nonempty ideal in $D$. I have to show that there are only finitely many ideals $J$ in $D$ such that $I$ is contained in $J$. My first idea would be: assume that ...
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### Prove that $D(\alpha)=D(\beta)$

Let K be an algebraic number field. Let $\alpha \in$ K. Let $\beta$ be conjugate of $\alpha$ relative to K . Prove that $D(\alpha)=D(\beta)$. $D(\alpha)$:= Let K be algebraic number field of degree ...
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### Ireland and Rosen, question 12.28 (unique factorization in prime ideals)

I'm stuck with the following exercise in Ireland and Rosen, chapter 12. Let $D$ be the ring of integers in a number field $F$. Suppose $(p)=P^2A$ for $p$ prime in $\Bbb Z$ and a prime ideal $P$. ...
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### If $\alpha$ and $\beta$ are algebraic integers then the roots of $x^2+\alpha x+\beta$ are algebraic integers

(This question is a dupplicate from If $\alpha$ and $\beta$ are algebraic integers then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.) I'm trying to solve this problem with ...
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### The field of algebraic numbers in $\mathbb Q (a_1,\ldots, a_l)$ is finite over $\mathbb Q$

In the book 'Algebra IV: Infinite Groups, Linear Groups' by Kostrikin and Shafarevich, there is a sketch of a proof (on page 84) of a theorem by Schur. I'm struggling to understand the line: Since ...
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### $p$ is an odd prime of the form $p=x^2+2y^2$ iff $p\equiv_8$ $1$ or $3$ [duplicate]

How would I prove the following: Show that an odd prime $p$ can be written on the form $p=x^2+2y^2$ for some $x,y\in\mathbb Z$ iff $p\equiv_8 1, 3$. Hint: use the quadratic reciprocity and the ...
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### How many prime ideals does $\mathbb Q[x]/(x^m -1)$ have? (multiple choice)

Let $m$ be a positive integer, and $a_m$ denote number of distinct prime ideals of $\mathbb Q[x]/(x^m -1)$. Then which of the following are true? $a_4=2$ $a_4=3$ $a_5=2$ $a_5=3$