# Questions tagged [algebraic-number-theory]

Questions related to the algebraic structure of algebraic integers

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### Let $D,m$ be relatively prime integers with $m$ odd. Then $D \equiv 0,1 \pmod 4$ and $D \equiv b^2 \pmod m$ implies that $D \equiv b^2 \pmod{4m}$

This is from a proof in David A. Cox's Primes of the Form $x^2+ny^2$: Lemma 2.5 Let $D\equiv 0,1 \bmod 4$ be an integer and $m$ be an odd integer relatively prime to $D$. Then $m$ is properly ...
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### What does the notation "$(R, +, \cdot)$" and "$(C, +, \cdot)$" mean? (From a book on Number Systems.)

The number "$i$" also is as much of a mental construct and no more , as the number "$s$". The new number is defined in such a way that it not only satisfies $i^2=-1$, but when ...
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### ramified in prime degree cyclic extension implies totally ramified in prime power cyclic extension

Let $K/\mathbb Q$ be a prime power cyclic extension, say of degree $p^n$. If a prime $q$ ramifies in the subfield $K_1$ of $K$ such that $[K_1:\mathbb Q]=p$, can one assert that $q$ is totally ...
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### What am I misunderstanding in the definition of this subgroup?

Let $K$ be a number field with $d \in \mathcal{O}_K \setminus \{0\}$, $\sqrt{d} \notin \mathcal{O}_K$. Consider the group of units $\mathcal{O}_K[\sqrt{d}]^\times$ and denote conjugation of elements ...
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### Tate gamma factor as a principal value integral

Let $F$ be a local field, $\chi$ a multiplicative character of $F^{\ast}$, and $\psi$ an additive character of $F$. The gamma factor $\gamma(s,\chi,\psi)$ is defined by means of the local functional ...
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### van der Waerden's proof that a monic $p(x) \in \mathbb{Z}[x]$ has Galois group $S_n$ with probability 1

This paper mentions that van der Waerden proved some results on the density of monic integer polynomials with Galois group the symmetric group $S_n$ in 1936. I have found van der Waerden's original ...
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### Can one show that in a certain sense, "most" polynomials have Galois group $S_n$? [duplicate]

Intuitively, it seems that given a random irreducible $p(x) \in \mathbb{Q}[x]$ of degree $n$, the Galois group of $p(x)$ over $\mathbb{Q}$ should be $S_n$. Otherwise, if $\alpha_1,...,\alpha_n$ are ...
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### why $|a|=|\sigma (a)|$, where $|\cdot|$ denotes absolute value on $\overline{ \Bbb Q_p}$?

Let $a\in\overline{ \Bbb Q_p}$, $\sigma\in Gal$( $\overline{ \Bbb Q_p}/ \Bbb{Q}_p$）, then, why $|a|=|\sigma (a)|$, where $|\cdot|$ denotes absolute value on $\overline{ \Bbb Q_p}$? I think we should ...
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### What is the local basis at $0$ of inverse limit topology?

What is the local basis of inverse limit topology at $0$? For example, $\mathbb Z_p＝\lim\mathbb Z/p^n\mathbb Z$ has $$\{ \{(\cdots,α_0)\in\mathbb Z_p｜α_m＝・・・＝α_o＝0\} \mid m≧0\}$$as a local basis at $0$...
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### Totally ramified $\mathbb{Z}_p$ extension

Let $K$ be a $p$-adic field, that is, $K$ is of characteristic $0$ and the residue field is perfect of characteristic $p$. On many places, it writes let $K_\infty$ be a totally ramified $\mathbb{Z}_p$-...
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### Soft Question: Introductory books on Algebraic Number Theory

I am a high school student in Britain. I have recently studied elementary number theory and absolutely loved it. I got as far as to prove results like quadratic reciprocity, Fermat's sum of two ...
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### Why $p$-adic logarithm is continuous on $\mathbb{Q}_p^\times$?

Neukirch defined, in Algebraic Number Theory, Neukirch (5.4) p. 136, the $p$-adic logarithm on $1 + p\mathbb{Z}_p$ as $\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$ Then, he extends this ...
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### Does $\Bbb{Q}_2$ has $\sqrt{-1}$?
Does $\Bbb{Q}_2$ has $\sqrt{-1}$? I tried to use Hensel lemma as usual. Let $f(x)＝x^2＋1$. But if some $a∈\Bbb{Z}$, $f(a)＝0$, then $f'(a)$ can always divide by $2$. So I cannot use Hensel lemma. Could ...