# Tag Info

## Hot answers tagged algebraic-number-theory

### How to determine multiplicative dependence of two algebraic numbers?

Here is a general algorithm, although applying it to your case seems laborious. We can work in the number field $K$ generated by $x$ and $y$. The first thing we can try is a generalization of your ...
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### Calculate the ideal class group of $K=\mathbb{Q}(\sqrt[3]{11})$

By your previous analysis, $\alpha-1 \in \mathfrak{p}_2\cap\mathfrak{p}_5 = \mathfrak{p}_2\mathfrak{p}_5$, so there is an ideal $I$ such that $\mathfrak{p}_2\mathfrak{p}_5I=(\alpha-1)$. By computing ...
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### Rational matrix whose power is an integer matrix

Since the characteristic polynomial has integer coefficients, there are matrices $B \in M_n(\mathbb{Z})$ and $P \in GL_n(\mathbb{Q})$ with $A = PBP^{-1}$ (by the MO answer linked in the question). Let ...
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### How to determine multiplicative dependence of two algebraic numbers?

For the particular example, this might help: $x^4= -\frac{3}{5} + \frac{4}{5}i$. Now if we could show that $y^n\not \in \mathbb{Q}(i)$ for all $n>0$. That probably follows from some basic field ...
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### Proposition 2.6(b) in The Arithmetic of Elliptic Curves - fiber cardinality is almost always equal to separable degree for a map of curves

Yes, your final paragraph is almost correct. A purely inseparable morphism of schemes which is also projective with integral source and integral, normal, locally noetherian target has geometrically ...
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### Find the ideal class group of $\mathbb{Q}(\sqrt{-5})$ by using the factorization theorem

Let me try to answer your questions. If there are any questions left, please let me know. First off, I'll answer 2. The definition of a "prime ideal" is different than what you might think, ...
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### The integral basis of $\mathbb{Q}(\zeta+\zeta^{-1})$

First, a priori you want to show that every element of $\mathcal{O}_K$ is of the form $\sum_{i=0}^{\frac{p-3}{2}} t_i (\zeta+\zeta^{-1})^i$ with $t_i\in\mathbb{Z}$, and not of the form \$t_0+ \sum_{i=1}...

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