Questions tagged [algebraic-integers]
For questions regarding algebraic integers, which is a complex number which is integral over the integers.
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Decomposition of rational primes, why work over ring of algebraic integers?
The proof of Thm. 6.1 in this article https://people.reed.edu/~jerry/361/lectures/iqclassno.pdf actually proves the following generalization of Thm. 6.1 beyond quadratic number fields (copied from the ...
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Why Does (3) Completely Ramify in $\mathbb{Q}(\omega, \sqrt[3]{2})$?
I'm working through this notes on Algebraic Number Theory. In section 2.6 they claim (3) is completely ramified over the larger field:
If we know that the ring of integers of $\mathbb{Q}(\omega, \sqrt[...
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Proof that $\frac{V_n(k_1,...,k_n)}{V_n(1,...,n)}$ is a integer.
Let $k_1 < k_2 < ... < k_n$ will be integers. Prove that the quotient
$\frac{V_n(k_1,...,k_n)}{V_n(1,...,n)}$ is a integer.
where $V_n(x_1,...,x_n)$ is vandermonde determinant
My idea: Show ...
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Is there an integral extension of $\mathbb{Z}$ keeping its prime elements?
Let $R$ be a ring. An extension $S$ of $R$ is called prime-keeping if every element $p$ prime in $R$ is also prime in $S$.
Consider the ring $\mathbb{Z}$ and the following two extensions: $\mathbb{Z}[...
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For which $d$, the algebraic integers of $K =\Bbb{Q}(\sqrt d)$ is a PID?
I know that the matter is settled for $d<0$ and an open problem for $d>0$ but I am asking about already known values. The below theorems are the motivation for asking this question.
Let $d$ be ...
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Can every transcendental number be expressed as the infinite sum of a quotient of two polynomials?
Is it possible to express all transcendental numbers (and more generally all real numbers $\in \mathbb R$) as the sum of an infnite series of the form
$$\sum_{n=0}^{∞} \frac{p(n)}{q(n)}$$ where $p(n)$ ...
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Symmetric Expressions in Roots of Polynomial with Coefficients in Integral Domain
Recently I learned about Pisot-Vijayaraghavan numbers, and proofs around them led me to the fact that the sums of $n$th powers of roots of a monic irreducible integer polynomial are integers. I saw a ...
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Proof of Dedekind-Kummer theorem
I will write the statement and then ask my query about part of the proof.
Statement
Let $p$ be a rational prime. Let $K=\mathbb{Q}(\theta ) $ be a number field where $\theta $ is an algebraic integer. ...
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Proving quotient of $\mathcal{O}_K $ with ideal $I$ is finite.
I need help showing the following
:
Suppose $K$ is a number field of degree $n$ and $\mathfrak{a} $ is a non-zero ideal of $\mathcal{O}_K $.
Then $\mathfrak{a} $ as an additive finitely generated ...
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Algebraic number theory - finiteness of quotient
In part of a proof in reading that proves $\mathcal{O}_K / \mathfrak{a} $ is finite for any nonzero ideal $\mathfrak{a} $ of $\mathcal{O}_K$.
It says that since $\mathcal{O}_K$ is a finitely generated ...
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Algebraic number theory proof question
Let $K$ be a number field of degree $n$ and let $H$ be a finitely generated subgroup of $\mathcal{O}_K^+$ of rank $n$.
Then $H$ has a $\mathbb{Z}$-basis $\omega_1,...,\omega_n $.
$\Delta (\omega_1,...,...
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