Tagged Questions

Questions related to the algebraic structure of algebraic integers

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Number of square-free polynomials over a finite field - a combinatorial interpretation?

One can show using zeta functions that the number of (monic)square-free polynomials of degree $n$ over a finite field $\Bbb F_q$ is $q^n-q^{n-1}$. For instance, this is done here. The answer is ...
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Show that the prime ideals above a prime $p$ are principal

Let $K=\mathbb Q(\alpha)$ where $\alpha^3 -5\alpha + 5 = 0$. I need to show that the prime ideals above 5 are principal, and find a generator for them. I have worked out the prime decomposition of ...
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How to tell if a set is open in the Krull topology?

I'm an undergraduate not very familiar with topology trying to understand the so called Krull Topology in the context of infinite Galois Theory. We proceed as follows: Let $\Omega/k$ be a (possibly ...
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Irreducibles elements in $\mathbf Z[\sqrt{-3}]$

The ring $A:=\mathbb Z[\sqrt{-3}]$ is the prototype of the rings usually used in a first algebraic number theory course to show the difference between prime and irreducible elements. I was wondering ...
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“Lifting” fibres of morphism of arithmetic schemes to get rid of “nongeometric” ramification

This is a soft question and really a request for pointers towards a certain rigorous formulation of geometric intuition I've had for some "arithmetic schemes". I'm looking for ideas and key references ...
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Ramification of primes in cyclotomic field

I don't know a lot of algebraic number theory, but I think the following is true: let $E/\mathbf{Q}$ be an algebraic number field, and $p\in\mathbf{Z}$ a prime. Then there a unique integers $e,f,g$ ...
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Self Learning — Number Theory

I was wondering if there were any good online courses/lecture videos (preferably courses/videos but books would work too) for self learning algebraic number theory. I have seen sites like MIT ...
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Compute the decomposition of $5\mathbb{Z}_K$ as a product of prime ideals

Let $K = \mathbb{Q}(\alpha)$ such that $\alpha^3 - 5\alpha + 5 = 0$. It is easy to show that $\mathbb{Z}_K = \mathbb{Z}[\alpha]$ and that $5$ is not maximal in $\mathbb{Z}[\alpha]$. So we cannot use ...
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Showing $A[\theta] \subseteq B \subseteq \frac{1}{d} A[\theta]$, where $A$ is a Dedekind domain

Suppose we have $A$ a Dedekind domain, $K$ its field of fractions. Let $L$ be a field extension of $K$ of degree $n$ and also that it is separable. Let $B$ be the integral closure of $A$ in $L$. ...
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Showing $B_P$ is a finitely generated module over $A_P$ where $P$ is a prime ideal in a Dedekind domain.

Suppose we have $A$ a Dedekind domain, $K$ its field of fractions. Let $L$ be a field extension of $K$ of degree $n$ and also that it is separable. Let $B$ be the integral closure of $A$ in $L$. ...
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Completion of an extension $K|\mathbb{Q}$ w.r.t. a non archimedean abs. value, isomorphic to $K\cdot \mathbb{Q}_p$

As the title suggests I want to prove the following result: given $\mathfrak{p}|(p)$, prime ideal of $\mathcal{O}_K$ over $p$, we have the canonical absolute value induced by it on $K$, and we can ...
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Prime $\mathfrak{p} \in$ Max$(\mathbb{Z}[\sqrt{10}])$ splits completely iff principal

L.S., This is an exercise from my lecture notes on algebraic number theory: Let $L = \mathbb{Q}(\sqrt{2},\sqrt{5})$ and $K = \mathbb{Q}[\sqrt{10}]$. Prove that prime ideal $\mathfrak{p} \in$ ...
The sum of all primitive $n$-th roots of unity in the algebraic closure of $F(x)$
If $n$ is any natural number, then $\mu(n)$ is the sum of all primitive $n$-th roots of unity in $\mathbb{C}$ where $\mu$ is the Möbius function defined as $\mu(n)=0$ if $n$ is not square free and ...
Completion of a Galois Extension $K | \mathbb{Q}$ w.r.t. to a non archimedean abs. value is a Galois extension
As the title suggests, I'm interested in a proof of the following fact: Let $K | \mathbb{Q}$ be a finite Galois extension, and let $\mathfrak{p}$ be a prime ideal of the ring of integers of $K$. ...