# Tagged Questions

Questions related to the algebraic structure of algebraic integers

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### Properties of the norm in a Euclidean Domain

I am aware of the fact that the Euclidean Norm does not need to be unique in a given domain, however my question is essentially: can we ensure that the properties of the norm remain the same? More ...
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### What are the units in $\mathbb{Z}[\root 3 \of 2]$?

I asked Wolfram Alpha to tell me the fundamental unit of $\mathbb{Z}[\root 3 \of 2]$, it replied $1 - \root 3 \of 2$. Then I tried asking it for $(1 - \root 3 \of 2)^n$ for $-5 \leq n \leq 5$. If I ...
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### Transitivity-like Results in Group, Ring, Module, Field and Galois Theory [closed]

I am reading Michael Atiyah and Ian Macdonald's Introduction to Commutative Algebra. On page 28, Proposition 2.16 says: Suppose $A,B$ are rings, $N$ is a finitely generated $B$-module, $B$ is ...
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### Show that $\xi^3\equiv \pm 1 \pmod{\lambda^4}$ in $\Bbb Z [\omega]$

We have $\lambda=1-\omega$ where $\omega=e^{i 2\pi/3}$ and $\xi$ an Eisenstein integer. Given that $\xi \equiv \pm 1 \pmod{\lambda}$, how can I prove that $$\xi^3\equiv \pm 1 \pmod{\lambda^4}$$ I ...
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### An problem of ideal splitting in number field extension

If $L/K$ is Galois extension of number field, $\mathfrak{p}$ is an prime integral ideal of $K$. One would asserts that: $\mathfrak{p}\mathcal{O}_L=\mathfrak{P}_1^{e_1}\dots\mathfrak{P}_g^{e_g}$. ...
Consider we have a stricktly increasing positive sequence $\lambda_n$ and the following sixth order algebraic equation for every $n\in \mathbb{N}$, $$\zeta s^6-s^4+\lambda_n^2=0,$$ where $\zeta$ is a ...
Kronecker's theorem says that a field extension can be shown as say, F(a) represented as F[x]/minimalpoly(a). Say, Q[$\sqrt{2}$]=Q[x]/$(x^{2}-2$) And a well known example is Q[$\sqrt{2}+\sqrt{3}$]=Q[...