Questions related to the algebraic structure of algebraic integers

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0
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3answers
99 views

Showing reducibility of a polynomial in a Discrete Valuation Ring

Let $R$ be a complete discrete valuation ring with uniformiser $\pi$. I would like to show that a polynomial $f$ in $R[X]$ is reducible. Does it suffice to show that $f$ is reducible in $\frac{R}{\pi^...
2
votes
2answers
33 views

ramification index in an example

Let $L=\mathbb{Q}_5[x]/(x^4+5x^2+5)$, where $\mathbb{Q}_5$ is the field of 5-adic numbers. Note that the polynomial that we are quotienting out by is an Eisenstein polynomial. So $L/\mathbb{Q}_5$ is a ...
0
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1answer
26 views

Neukirch Abstract Kummer Theory. Understanding a Proof.

This question is a sort of follow up to this question, where I introduced context. Neukirch mysterious homomorphism in Abstract Kummer Theory (in his book ANT) The thing I don't understand know is ...
0
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0answers
36 views

$p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
2
votes
1answer
53 views

Algebraic number theory, Marcus, Chapter 3, Question 9

Question 9 in Marcus book. Let $K$ and $L$ be the number field such that $K\subset L$ and let $R,S$ be their algebraic integers, respectively. a) Let $I$ and $J$ be ideals in $R$, and suppose $IS|...
1
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3answers
41 views

Norm of element $\alpha$ equal to absolute norm of principal ideal $(\alpha)$

Let $K$ be a number field, $A$ its ring of integers, $N_{K / \mathbf{Q}}$ the usual field norm, and $N$ the absolute norm of the ideals in $A$. In some textbooks on algebraic number theory I have ...
0
votes
1answer
53 views

Neukirch mysterious homomorphism in Abstract Kummer Theory (in his book ANT)

Someone familiar with Neukirch's terminology can understand this post better. Unfortunately it is so much terminology to just explain it here. My question is about what is marked in the picture: Why ...
1
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1answer
23 views

Algebraic number with conjugates having modulus 1

Suppose $\alpha$ is an algebraic number lying in a number field $K$ that is a normal extension of $\mathbb{Q}$. Suppose all the conjugates of $\alpha$ have absolute value 1. Prove or disprove that it ...
3
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1answer
56 views

Computing the Picard group of Number Fields in analogy with genus $0$ curves

It is possible to compute the Picard group of (some? all?) genus $0$ curves in the following manner: For concreteness let, $A = k[x,y]/(x^2+y^2-1)$ and $X = \operatorname{Spec} A$. Let $Y$ denote $\...
6
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1answer
118 views

Minimal polynomial of root of unity over quadratic field

Let $p$ be an odd prime and consider the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$ and its quadratic subfield $\mathbb{Q}(\sqrt{\pm p})=:K$. I am interested in the minimal polynomial of a root of ...
2
votes
1answer
98 views

Semilinear root of uniformiser of a p-adic field (& phi-module of Lubin–Tate formal group)

I'm looking for solutions $t$ of an equation of the form $$ t \sigma(t) \cdots \sigma^{n-1}(t) = v $$ in a field equipped with an automorphism $\sigma$ of order $n$. In this case, I call $t$ a "$\...
3
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0answers
57 views

Enumeration of the number of splitting fields

Suppose $f(x):=x^p+ax+b\in \mathbb Z[x]$ and let $S_f$ be the minimal splitting field of $f(x)$. How can we estimate $\#\{(a,b):|S_f:\mathbb Q|=2p\}$?
3
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0answers
59 views

“Logarithmic derivative of a p-adic number”

Coleman, in "Division Values in Local Fields", (Inventiones math. 53, 91 - 116 (1979)), says that In his work on cyclotomic fields Kummer observed that various formal operations on power series ...
15
votes
2answers
264 views

Primes of form $a^2 + 24b^2$

For a prime number $p \neq 2$, $3$, is it necessarily the case the prime number can be written in the form $a^2 + 24b^2$ if and only if $p \equiv 1 \text{ mod }24$? I think this has to be true based ...
1
vote
1answer
60 views

Absolute Galois group of quadratric extensions

It is well known that the absolute Galois group of $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{q})$ are nonisomorphic if $p$ and $q$ are different prime numbers. See for example Szamuely's book "...
1
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0answers
19 views

Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
2
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1answer
42 views

Neukirch's Abstract CFT. Help with a proof in abstract Kummer theory.

First of all, unfortunately, writing all the notation and terminology that he uses would make this post very big. So, I'm really hoping from an answer that comes from someone that knows this book. ...
2
votes
1answer
55 views

Algebraic Number Theory,Marcus, Chapter 2, Question 16

In question 16 of chapter 2 in Marcus Book, I have to show that $\sqrt{3}\not\in\mathbb{Q}(\alpha)$,where $\alpha=\sqrt[4]{2}$ using the trace idea. the proof starts by assuming that $\sqrt{3}=a+b\...
3
votes
1answer
75 views

$p = x^2 + xy + 3y^2$ if and only if $p \equiv 1$, $3$, $4$, $5$, $9$ mod $11$? [duplicate]

For a prime number $p \neq 11$, do we have $p = x^2 + xy + 3y^2$ for some $x$, $y \in \mathbb{Z}$ if and only if $p \equiv 1, 3, 4, 5, 9$ mod $11$? An example where this is true:$$5 = 1^2 + 1 \times ...
1
vote
0answers
30 views

Roots of unity in fixed field of decomposition group.

$\zeta_q\in L^{G({\frak P})}\Leftrightarrow$ if $\sigma\in$ $G(\frak p$) then $\sigma(\zeta_q)=\zeta_q\Leftrightarrow$ if $\sigma$ fixes $\frak P$ then $\sigma$ fixes $\zeta_q$. What's next? Suppose ...
1
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0answers
64 views

Probability that the discriminant of a quadratic field is divisible by a given prime number $p.$

Find the probability that the discriminant $D$ of a quadratic field $\mathbb{Q}(\sqrt{d})$ is divisible by a given prime number $p$ (beware: the result is not what you may expect.) This is an ...
12
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9answers
215 views

$p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?

For a prime number $p \neq 3$, do we have that$$p = x^2 + xy + y^2$$for some $x$, $y \in \mathbb{Z}$ if and only if$$p \equiv 1 \text{ mod }3?$$I suspect this is true from looking at the example$$7 = ...
3
votes
2answers
53 views

What does $p$-integral mean?

I'm currently studying Washington's Introduction to Cyclotomic fields and in Theorem 5.10 I came across the term $p$-integral. What does this mean? To give a bit of context: Let $n$ be even and ...
2
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0answers
58 views

Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
0
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0answers
22 views

Abstract CFT. What does Neukirch mean with exponent here?

In Neukirch's Algebraic Number Theory, there is the following Proposition: If we take $K$ a field with characteristic $5$, $n=4$, and choose $\Delta=K^{\ast\:4}$, wouldn't it imply that the trivial ...
0
votes
1answer
32 views

Galois group of a subextension is a subgroup of the Galois group of the extension?

Let $\overline{\mathbb{F}}_{5}$ be the separable closure of $\mathbb{F}_{5}$ and let $G=G(\overline{\mathbb{F}}_{5}/\mathbb{F}_{5})$ be its Galois group. Say we pick a finite subextension of $\...
2
votes
1answer
78 views

What's the sense behind that lemma?

Please if someone can help and can take 3 minutes I would be so so unbelievably happy because it is really important to me... Thank you :) We assume we have a $m$-th root of unity $\zeta_m=e^{\frac{2\...
2
votes
1answer
35 views

Ramification of extension composition

$L, E\supset K$ are number fields. $L/K$ is normal. And field $M=LE$. Assume $\Omega$ is a prime ideal of M and its intersections with $L, E, K$ are $\mathfrak B,\mathfrak q,\mathfrak p$. $(1)$ ...
2
votes
1answer
57 views

Proof of Kummer's Lemma in S. Langs 'Cyclotomic fields'

I was going through the proof of Kummer's Lemma (stated below) as done in Serge Langs Cyclotomic fields on page 312. Now the author states that by class field theory it suffices to show that $\...
2
votes
3answers
40 views

Extension of finite fields under norm map

If $L/K$ is a finite Galois extension with group $G$, we can define the norm of an element $a\in L$ as \begin{equation} N_{L/K}(a)=\prod_{\sigma\in G}\sigma(a). \end{equation} And obviously, this ...
4
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0answers
36 views

Given only the minimal polynomial of a field and approx primitive element, find subfields

I want to solve the following problem. Given: The minimal polynomial of a number field, and a decimal approximation of the field's primitive element. Compute: Is the number field an imaginary ...
2
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1answer
46 views

When do imaginary quadratic extensions have unique complex places?

This question has been edited in light of some helpful comments from @AdamHughes below. Let $F$ be a totally real number field, i.e. $F=\mathbb{Q}(t)/m(t)$ where $m(t)\in\mathbb{Z}[t]$ and all roots ...
3
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1answer
30 views

Decomposition and inertial fields of primes in $\mathbf{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$

I recently ran into this old number theory prelim problem. Let $K=\mathbf{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ and let $\mathcal{O}_K$ be the ring of integers of $K$. Find the ramification index and ...
1
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1answer
43 views

Problem from the book Number Fields by Marcus

I have been stuck on the 14(c)th problem of the 3rd chapter from Marcus' Number Fields. Let $K$ and $L$ be number fields, $K \subset L$, $R = \mathbb{A}\cap K$, $S = \mathbb{A} \cap L$. Moreover ...
7
votes
3answers
194 views

Does $p = x^2 + 9y^2$ for some $x$, $y \in \mathbb{Z}$ if and only if $p \equiv 1 \text{ mod }12$?

For a prime number $p \neq 2$, $3$, does $p = x^2 + 9y^2$ for some $x$, $y \in \mathbb{Z}$ if and only if $p \equiv 1 \text{ mod }12$? A case where this is true as to suggest plausibility: $13 = 2^2 +...
0
votes
1answer
23 views

Taking $d$-th root of an element in the algebraic closure of $\mathbb{Q}$

Suppose I have $\alpha \in \overline{\mathbb{Q}}$, the algebraic closure of $\mathbb{Q}$. Suppose I was interested in an element $\beta$ such that $\beta^d = \alpha$. Does there always exist $\beta \...
3
votes
0answers
54 views

Infinite primes (places) of a number field geometrically

Given a (global) number field $K$, thinking of the affine scheme $\mathrm{Spec}\mathcal{O}_K$ can gige an insight into (at least) some kf the number-theoretic terminology, e.g. ramification or local ...
3
votes
1answer
69 views

Completions of number fields

I would like to prove a statement about completions of number fields, but I'm running into a problem. The statement I want to prove is Let $L/K$ be a Galois extension of number fields, $p$ a prime ...
5
votes
1answer
108 views

A prime ideal $\mathfrak{p}$ decomposes in $\mathbb{Q}(\zeta_{12})/\mathbb{Q}(i)$ iff it is generated by $\alpha\in1+3\Bbb{Z}[i]$

Prove that for a nonzero prime ideal $\mathfrak{p}$ of $\mathbb{Z}[i]$ which does not divide $3$, $\mathfrak{p}$ decomposes completely in the quadratic extension $\mathbb{Q}(\zeta_{12})/\mathbb{Q}(i)$ ...
10
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1answer
281 views

Three angles are linearly independent over $\mathbb{Q}$?

If$$\tan \alpha = 1, \text{ }\tan \beta = {3\over 2}, \text{ }\tan \gamma = 2,$$then does it follow that $\alpha$, $\beta$, $\gamma$ are linearly independent over $\mathbb{Q}$? It is possible to test ...
9
votes
4answers
110 views

Subgroup generated by $1 - \sqrt{2}$, $2 - \sqrt{3}$, $\sqrt{3} - \sqrt{2}$

For a number field $K$, Dirichlet's unit theorem says that$$(O_K)^\times = \mathbb{Z}^{r - 1} \oplus (\text{a finite cyclic group}),$$where $r$ is the number of all infinite places of $K$. An infinite ...
1
vote
1answer
19 views

irreducible in the ring of integers

There is a primitive 12-th root of unity and 5 is not a prime since the minimal polynomial mod 5 is reducible. The problem is I don't know how to show 5 is irreducible or not. What I thought was if ...
2
votes
0answers
53 views

Is $(2, \sqrt{m})$ a principal ideal or not in the ring $\mathbb{Z}[\sqrt{m}]$ [closed]

Let $m$ be a negative even integer. In the ring $\mathbb{Z}[\sqrt{m}]$, is the ideal $(2,\sqrt{m})$ a principal ideal?
3
votes
1answer
52 views

Class number of $\mathbb{Q}(\sqrt{n})$ always even? [closed]

Let $n$ be a negative square-free even integer. Does it necessarily follow that the class number of $\mathbb{Q}(\sqrt{n})$ is even?
5
votes
1answer
50 views

Element of $\mathbb{F}_{p^2}$ of order $p^2$ raised to $p+1$th power is element of $\mathbb{F}_p$?

For any prime number $p$ and any element $a$ of the finite field $\mathbb{F}_{p^2}$ of order $p^2$, do we have$$a^{p+1} \in \mathbb{F}_p \subset \mathbb{F}_{p^2}?$$
2
votes
2answers
67 views

What does algebraic number look like locally?

Is there any theorem characterizing what algebraic number looks like locally (in completion)? For example, do all algebraic numbers live in some $\mathbb{Q}_p$? Does there exist algebraic number in ...
2
votes
1answer
46 views

Why must a zero of $f \in \mathbb{Z}_p[X_1, \dots, X_m] ~ (\text{mod } p^n)$ be simple in order to lift to $\mathbb{Z}_p$?

In chapter II, section 2.2, of J-P. Serre's A Course in Arithmetic, we have the following theorem: Theorem 1: Let $f \in \mathbb{Z}_p[X_1, \dots, X_m]$, $x = (x_i) \in \left( \mathbb{Z}_p \right)^...
3
votes
1answer
91 views

Examples of how to apply algebraic number theory

I am reading about algebraic number theory mainly following milne's notes. But currently I really wonder how such theory can help solve problems of number theory. One example I know is we can use ...
2
votes
1answer
86 views

Is 5 a prime element in the cyclotomic ring of integers?

Given a primitive 12-th root of unity, so its minimal polynomial is $$x^4-x^2+1$$ and hence the degree of its cyclotomic ring of integers is 4. Recently I've learnt about quadratic field and ring of ...
3
votes
1answer
66 views

Examples for abstract class field theory?

I'm starting to get into Abstract Class Field Theory, following Neukirch's famous ANT. The initial setup is basically a profinite group $G$ and a discrete abelian group $A$ on which $G$ acting as ...