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,...,\omega_n)=\text{Det} (c_{ij})_{n \times n} $ where $c_{ij}=\text{Trace}_{K/\mathbb{Q} }(\omega_...

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 additive abelian group we have $\mathcal{O}_K \cong \mathbb{Z}^n $ for some natural $n$. (Here w...

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 abelian subgroup of $\mathcal{O}_K^+ $ has rank $n$. Here is what I know $\mathcal{O}_K$ Is a finitel...

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. Suppose $p \nmid [\mathcal{O}_K : \mathbb{Z}[\theta ]] $. Let $$ \mu _{\theta }\equiv f_1^{e_1}....

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 proof, but it used power series, and I thought it must be provable with Galois Theory, so I came ...