14
votes
2answers
2k views

Is there an algebraic number that cannot be expressed using only elementary functions?

(this is basically a repost of a question I asked at M.SE last year) Is there an explicit real algebraic number (such that we can write its minimal polynomial and a rational isolating interval) that ...
6
votes
3answers
485 views

Constructing quintic number fields with certain splitting behaviour

I am looking for number fields $K$ which satisfy the following properties: $[K:\mathbb{Q}]=5$. The Galois closure of $K$ has Galois group $S_5$. For each prime $p$ which ramifies in $K$, there ...
25
votes
0answers
372 views

Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...
31
votes
1answer
875 views

Is my field algebraically closed?

For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having prime-power degree. Question: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$? ...
1
vote
0answers
85 views

Discriminant of a compositum of number fields, a bound?

Given two number fields $E$ and $F$, is there a bound on $|d_{EF}|$, the absolute value of the absolute discriminant of the compositum of fields $EF$, in terms of $d_E$, $d_F$, and the extension ...
6
votes
1answer
240 views

Degree of Kummer extensions of number fields

Let $K$ be a number field and $a\in K^*$ of infinite order in $K^*$. How do I show that $$[K(\sqrt[n]{a},\zeta_n):K]\geq C\cdot n\cdot\varphi(n)$$ holds for all positive integers $n$, with a positive ...
3
votes
1answer
216 views

Unramified extension and class field theory

I am not sure this question is proper for this site, but there is no other places that I can get an answer. So if anyone can give an answer for this, it would be very helpful to me. Let $F$ be a ...
1
vote
1answer
166 views

Deciding if the largest absolute value real root lies in a cyclotomic extension

Given an algebraic equation of degree $n$ of form: $$x^{n} - a_{n-1}x^{n-1} - a_{n-2}x^{n-2} - \dots - a_{0} = 0$$ where each $a_{i} \in \Bbb Q^{+}$ and atleast one positive root, how does one decide ...
3
votes
1answer
273 views

Langlands Paper on representations of abelian algebraic groups

I have been working through Langlands paper which you can see here http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf and I can understand why one of his maps is obvious and ...
2
votes
1answer
336 views

is there any bound on the absolute number of algebraic integer in terms of its degree?

If Z is a sum of t distinct roots of unity and |Z| is a rational integer, can someone find a bound on |Z| in terms of k=deg(Q(Z):Q))? Clearly we need to have distinct roots of unity otherwise this ...
3
votes
1answer
344 views

Proof of a Simple Converse in Algebraic Number Theory

If $L/K$ is a Galois extension, then any prime $\mathfrak{p}$ of $K$ splits into a product ${\mathfrak P}_1^e\cdots {\mathfrak P}_g^e$ of primes in $L$, and the exponents on the primes are equal since ...
6
votes
0answers
601 views

Automorphisms of local fields

It is an amusing coincidence (at least it appears to be a coincidence to me) that any completion of the field $\mathbb{Q}$ has trivial automorphism group as an abstract field, i.e. when ignoring the ...
4
votes
2answers
1k views

A family of polynomials with symmetric galois group

Consider the following family of polynomials in $K[x,y]$, where $K$ has characteristic zero: $f_n(x,y)=(x+y)^n+(x-1)y^n,$ for $n\geq 3$. I can prove that $f_n(x,y)$ has an irreducible factor of ...
5
votes
1answer
1k views

Motivation for the proof of Hilbert's Theorem 90

The proof of Hilbert's Theorem 90 about cyclic extensions goes like this: Let $\sigma$ be the generator of the Galois group of order $n$ and let $b$ have norm $1$, i.e. $b \sigma(b) \cdots ...
5
votes
1answer
685 views

Effective Chebotarev density results for arbitrary number fields

So let $f(x)\in\mathbf{Z}[x]$ be a monic polynomial of degree $d$ and let $K$ be the splitting field of $f$. Let us define the "heigt of $f$" $:=||f||$ to be the maximum of the abolute values of the ...
6
votes
1answer
873 views

What is a “best” transcendence basis for R/Q ?

It is easy to show, using the axiom of Zorn, that there exists a transcendence basis for $\mathbb{R}/\mathbb{Q}$, i.e. a set $S$, algebraically independent over $\mathbb{Q}$, such that ...
12
votes
3answers
1k views

Finitely generated Galois groups

It is well-known that for a given natural number $n$ there is only finite number of extensions of $\mathbb Q_p$ of degree $n$. This result appears in many introductory books on algebraic number ...
7
votes
3answers
1k views

Infinite simple Galois groups

Conjecturally, every finite group is the Galois group of some extension of the rationals. This question made me wonder what is known about infinite simple groups occurring as Galois groups. What ...
2
votes
3answers
2k views

What (permutation) groups can occur as galois groups of irreducible polynomials of degree n

I think the answers for the first few degrees ($n$) are: $n=2$, $S_2$ $n=3$, $S_3,A_3$ $n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$ ($K_4$ is the Klein four group) $n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ ...
8
votes
3answers
432 views

Congruences mod primes in Galois extensions

I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta_m$, $\zeta_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields ...
1
vote
0answers
290 views

Splitting of prime ideals in non-Dedekind domains?

This is a follow-up to this question. So that you don't have to flick back and forwards I'll briefly summarize: My original question was on how to prove that a polynomial $g(x)$ obtained from ...
2
votes
1answer
676 views

How do I visualize finite covers of curves over non-algebraically closed fields?

If $L$ is algebraically closed, fields of transcendence degree one over $L$ correspond to algebraic curves over $L$ up to birational equivalence, and finite extensions correspond to finite Galois ...
2
votes
2answers
703 views

algebraic numbers of degree 3 and 6, whose sum has degree 12

This question is related to Degree of sum of algebraic numbers. Forgive me if this is a dumb question, but are there two algebraic numbers $a$ and $b$ of degree $3$ and $6$ respectively, such that the ...
6
votes
2answers
1k views

Is there a notion of Galois extension for Z / p^2?

The above title is in fact a special case of what I want to ask. Certainly we have a well defined notion of Galois extension for $ \mathbb{Q}_p $. The intersections of these extensions to the ring ...
11
votes
1answer
412 views

What are Mean Values of Ideal Densities in Galois Extensions?

In an unfinished (and as of now unpublished) article intended for the encyclopedia of mathematics, Arnold Scholz wrote: "Classifying extensions according to the Galois group of their normal closure ...
13
votes
5answers
2k views

Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?

Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? I am going to be on the road soon, so pleas don't be offended if I don't respond quickly to a comment.