# Tagged Questions

Questions related to the algebraic structure of algebraic integers

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### Is $p^n\mathbb Z_p\cong \mathbb Z_p$ as additive groups?

Is it true that $p^n\mathbb{Z}_p\cong \mathbb Z_p$ as additive groups? Here $\mathbb Z_p$ is the ring of $p$-adic integers for $p$ prime and $n$ is any positive integer. Thanks
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### If $\,p\,$ is prime, is $\,p^n\mathbb{Z}_p=\mathbb{Z}_p\,$ for any positive integer $n$?

Is $\,p^n\mathbb{Z}_p=\mathbb{Z}_p\,$ for any positive integer $n\,?$ $\mathbb{Z}_p =$ ring of $p$-adic integers, $\,p$ prime. Thanks.
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### Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
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### Generalization of Kummer isomorphism?

Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ ...
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### A characterization of the module function on a locally compact division ring

References: Weil's Basic Number Theory(written as BNT). Bourbaki's Commutative Algebra(written as BCA). Let $K$ be a topological ring with an identity. Suppose every non-zero element of $K$ is ...
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### Values of virtual characters

Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of n-...
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### Isomorphism of DVR quotient with quotient of completion

I'm confused about the proof of Lemma 7.25 from J.S. Milne's course notes on algebraic number theory, available here. Here is the situation: $A$ is the valuation ring corresponding to some discrete ...
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### Questions about integral normal bases for subfields of cyclotomic fields

An element $\theta$ in a Galois extension $L$ of $\mathbb{Q}$ is said to give an integral normal basis if the ring of algebraic integers in $L$ is $\sum \mathbb{Z}\sigma(\theta)$ with $\sigma$ running ...
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### $L$-function of character in terms of other character

Let $F/K$ be a finite Galois field extension and $\varphi : \mathfrak{I}_K \rightarrow \mathbb{C}^{\times}$ a Hecke character of $K$.Define $\psi = \varphi \circ N_{F/K}$ as a Hecke character of $F$. ...
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### Harmonizing local and global definitions of discriminant ideal

I began skimming PLC's 8430 handout 5 (on chebotarev density and global class field theory), and there are two versions of "discriminant" presented. The set up is that $R$ is a Dedekind domain, $K$ ...
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### Reference for relation between class number of $\Bbb Q[\sqrt{-p}]$ and partial quotients of $\sqrt p$

So in Ireland and Rosen's, "Classical Introduction to Modern Number Theory", they mention the following incredible fact at the end of Chapter 13, section 1. Suppose $p \neq 3$ and $p \equiv 3 \pmod 4$ ...
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### Index of ideals in rings of integers in number fields

Let $R$ be the ring of integers of a number field, or more generally, a finite index subring of it, and let $P$ be a prime ideal of $R$. Is there exist a good bound for the index $[R:P^n]$ in terms of ...
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### Primes inert in quadratic field of class number one

Let $K = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field of class number one (i.e. every ideal in $\mathcal{O}_K$ is principal, i.e. $\mathcal{O}_K$ is a principal ideal domain). Let $d_K$ be ...
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### what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
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### Separable polynomial mod $\mathfrak{p}$

Let $K/L$ be a galois extension with $L = K(\alpha)$ and let $f$ be the minimal polynomial of $\alpha$ over $K$. Let $\mathfrak{p}$ be a prime in $\mathcal{O}_K$ and $p$ be the prime number above ...
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### Finding exercises in local fields, following Serre's book

I am reading Serre's "Local Fields". I would like to find more exercises to complement my study. So I am looking for, mabye, exercises from a course that followed this book. Do you know of such a ...
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Let $K = \mathbb{Q}(\sqrt{41})$, $\omega = \frac{1}{2}(1+\sqrt{41})$, and $\mathcal{O}_K= \mathbb{Z}[\omega ]$ be the ring of integers of $K$. Let $\alpha = 27 +10 \omega$ be the fundamental unit of $\... 4answers 213 views ### Is the number of irreducibles in any number field infinite? Are there infinitely many irreducibles in the ring of integers of any algebraic number field ? I tried to follow the same argument as we usually do for integers. Suppose there are finitely many ... 2answers 148 views ### Why do we only consider quadratic domains as Euclidean domains with squarefree integers? I have been reading "Introductory algebraic number theory" by Alaca and Williams, and in the opening chapters they use the quadratic domains$\mathbb{Z}+\mathbb{z}(\sqrt{m})$for non-square$m$and$\...
Let $D$ be a Dedeking domain, $\mathfrak{i}$ a nonzero ideal of $D$ and let $B=D/\mathfrak{i}$ be the quotient ring. Then $B$ is a noetherian ring, and every prime ideal of $B$ is maximal. I have ...