Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

learn more… | top users | synonyms

5
votes
1answer
114 views

History of the three “impossible” compass-and-straightedge problems

I'm preparing a presentation about constructible numbers and I wanted to know some of the history about it to motivate the topic. I wanted to know if the classical Greek problems (doubling the cube, ...
0
votes
2answers
35 views

Algebraic Extensions

I have the following question: there is this statement i can't understand: Let $A$ be an integral domain which is integrally closed ( in its field of franctions ) and let $K$ be its fraction field. ...
0
votes
0answers
20 views

independence of Galois conjugates

Let $L \supset K$ be a transcendental extension of fields, $K$ algebraically closed and let $y \in L'$ be algebraic over $L$, and let $\sigma(y)$ be its Galois conjugate for some $\sigma \in ...
4
votes
2answers
63 views

If $f(x)\in\mathbb{Q}[x]$ of degree $p$ and $\operatorname{Gal}(K/\mathbb{Q})$ has element of order $p$ then $f(x)$ is irreducible.

Let $f(x)\in\mathbb{Q}[x]$ , $p$ prime, $\deg f(x)=p$ and $G = \operatorname{Gal}(K/\mathbb{Q})$ has element of order $p$, where $K$ the the splitting field of $f(x)$ over $\mathbb{Q}$. Show that ...
1
vote
0answers
40 views

Finite characteristic splitting fields of low degree polynomials

Let $f(x) \in \mathbb{Z}[x]$ be a polynomial. Let $f_p(x) \in \mathbb{Z}_p[x]$ denote the polynomial $f \bmod{p}$ (where $p$ is a prime). We say $p$ is good for $f$ if $f_p(x)$ splits (into linear ...
0
votes
2answers
40 views

If $\deg(f) > p^k$ then $f$ as an irreducible divisor of degree $> k$

Let $p$ be prime and let $f \in \mathbb{F}_p[X]$ with no repeated roots. Let $k \in \mathbb{N}^*$ such that $\deg(f) > p^k$. Show that $f$ has an irreducible divisor of degree $> k$. My ...
1
vote
1answer
39 views

What is the difference between the algebraic function fields and the fields itself

I'm studying this book and I don't understand exactly what's the difference between the algebraic function field $F/K$ and $F$ itself. Thanks
0
votes
1answer
49 views

Sum of roots of unity an algebraic integer proof

Let S be the sum of a finite number of nth roots of unity (where n is fixed, and the sum is non-zero). How do I go about showing that S is an algebraic integer in the cyclotomic field of order n ?
1
vote
1answer
55 views

Field extension of degree 3 and polynomial roots

Deleted the old question, because tho whole question kind of changed. I am facing following problem: Given extension of finite fields $L/K$ of degree $3$, prove that every polynomial of degree $3$ ...
6
votes
0answers
100 views

How often are Galois groups equal to $S_n$?

Let $\mathbb{Z}[x]_n$ be the set of polynomials in $\mathbb{Z}[x]$ of degree at most $n$. Then, consider some sensible increasing filtration $$A_0 \subset A_1 \subset A_2 \subset \cdots$$ of ...
0
votes
0answers
55 views

Relation between Galois theory and Fermat primes

I am curious about a possible relation between Galois theory and Fermat primes. There is a general solution to any polynomial equation of degree less than or equal to $4$. The only Fermat primes (of ...
2
votes
1answer
61 views

Decomposition group of a prime ideal and root of polynomials

Let $f(x)$ be a monic irreducible polynomial with integer coefficient. Let $K$ be the splitting field of $f$ and $\alpha$ one of its roots. Let $p$ a prime number such that $p$ does not divide ...
0
votes
2answers
51 views

every irreducible polynomial has a root in some field extension

We know the following fact from field theory. Let $F$ be a field and $p(X)$ an irreducible polynomial in $F[X]$. Then we can find a field extension $L$ of $F$ such that $p(X)$ has a root in $L$. ...
4
votes
1answer
55 views

Abelian Kummer Extension

A field extension of the form $\mathbb{Q}(\zeta_n, \sqrt[n]{\beta})$ where $\zeta_n$ is a primitive $n$th root of unity and $\beta \in \mathbb{Q}(\zeta)$ is called a Kummer extension. Even though ...
0
votes
2answers
56 views

dimension of $\mathbb{Q}(\sqrt2)$

How can I prove that the splitting Field $\mathbb{Q}(\sqrt2)$ over the rational numbers $\mathbb{Q}$ is two dimensional vector space over $\mathbb{Q}$ ?
0
votes
1answer
32 views

Constructing common extension of two finite fields

I am facing following problem, and would be very thankful for any input: I am supposed to prove that two finite fields with same number of elements are isomorphic. I know the usual proof via ...
5
votes
1answer
40 views

How to write a particular fixed field as a simple extension of $\mathbb{R}$? (Morandi's book)

I'm working in the following problem from Morandi's Book Field and Galois Theory: Let $A$ =$\left(\begin{array}{cc} a & b \\ c & d \end{array} \right)$ with $a=d=-1/2$ and ...
2
votes
1answer
39 views

Normal extension and group of automorphisms

Let $K \supset F$ a normal extension (finite). Show in details that, for every $\alpha \in K$, the polynomial $$f(x)=\prod_{\sigma \in Aut_F(K)}(x-\sigma(\alpha))\in F[x];$$ My attempt: Since $K$ ...
1
vote
0answers
102 views

Any general “formula” solutions for higher order polynomial equation?

We know that fifth (or higher) degree polynomial equation has no general solution formula using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of ...
4
votes
3answers
65 views

$[F(t):F(t^n)]=n$ where $t$ is trascendental

Let $F$ be a field and let $t$ be trascendental over $F$. Prove that $[F(t):F(t^n)]=n$. Obviously $[F(t):F(t^n)]\le n$ since the polynomial $f(x)=x^n-t^n \in F(t^n)$ has $t$ as a root. But I don't ...
1
vote
1answer
22 views

How to show that any solvable transitive subgroup of S$_p$ where $p$ is a prime has a conjugate contained in Aff($\mathbf F_p$)?

Here Aff ($\mathbf F_p$) denotes the group of affine transformations $x\rightarrow ax+b,$ with $ a\neq 0, b\in \mathbf F_p$. What I've done is to show that the penultimate group in the solvable series ...
1
vote
1answer
40 views

Prove that there exists a sequence of intermediate fields.

How can I prove that for $K\subset L$ - the Galois extension of degree $p^n$, where $p$ is prime, there exists a sequence $K=K_{0}\subset K_{1}\subset \cdots \subset K_{n}=L$ such that ...
4
votes
2answers
78 views

The determination of the Galois group of a polynomial

The GAP package has a function $\mathtt {GaloisType}$ that takes a polynomial as an argument and returns a number, the index of the transitive group of order the degree of the polynomial. I read ...
2
votes
1answer
52 views

Is $F\subset F(a,b)$ a Galois extension?

I know that $F\subset F(a)$ and $F\subset F(b)$ are the Galois extensions of degree $n$ and $m$ respectively, $(n,m)=1$. 1) How can I show that $F\subset F(a,b)$ is the Galois extension either? 2) ...
4
votes
2answers
80 views

Field Extension, Splitting Field and Galois Theory

Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n\geq 3$. Let $L$ be the splitting field of $f$, and let $\alpha \in L$ be a root of $f$. Given that $[L:\mathbb{Q}]=n!$, prove that ...
5
votes
1answer
63 views

Normal Basis Theorem Proof

I am a little confused by the proof of the Normal Basis Theorem in E. Artin's Galois Theory. Specifically, I am having trouble understanding why a certain squared matrix has a particular form. The ...
2
votes
1answer
55 views

Solvability of the groups.

Let $M$, $N$ be normal subgroups of a group $G$ such that $G/M$ and $G/N$ are solvable groups. How can I prove that $G/(M\cap N)$ and $G/\langle M,N\rangle$ are solvable either? Thanks in advance.
2
votes
2answers
70 views

Degree of the field extension.

Could you help me showing that for a field $F$ the degree of the extension $F(x^{2}+\frac{1}{x^{2}})\subset F(x)$ is $4$? I have found the polynomial $y^{4}-(x^{2}+\frac{1}{x^{2}})y^{2}+1$ such that ...
2
votes
0answers
62 views

Field made of real&imaginary parts of members of another field

Let ${\mathbb K}$ be a normal extension of $\mathbb Q$. Let $\mathbb L$ be the subfield of ${\mathbb C}$ generated by the real and imaginary parts of elements of $\mathbb K$. Thus $\mathbb L$ is a ...
1
vote
1answer
52 views

Calculating Splitting field

Find the splitting field of the polynomial and degree over $\mathbb{Q}$ $P(X)=X^4+2$. The roots of $P(X)$ are $\sqrt[4]{2}\sqrt{i},\ -\sqrt{i}\sqrt[4]{2}, \ i\sqrt{i}\sqrt[4]{2},\ ...
2
votes
1answer
34 views

Galois group of order 2^4

Find the galois group the polynomial $f(X)=(X^2-2)(X^2-3)(X^2-5)(X^2-7)$ over $\mathbb{Q}$. A splitting field for $f(X)$ is $K=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7})$. We must have ...
0
votes
1answer
23 views

Error on parity bits of Reed-Solomon error correction code

I'm trying to figure this out but it seems never to be covered in articles explaining Reed-Solomon codes. If I have a string with 64 characters (bytes) and 4 parity bytes for error checking and ...
0
votes
1answer
19 views

Terminology for Galois groups of non-Galois extensions.

I am having a confusion with terminology. If $L/K$ is not Galois, what is the meaning of "the Galois group of $L$ over $K$"? I have two guesses: 1) It is the field automorphisms of $L$ that fix ...
0
votes
2answers
48 views

Separable extension [closed]

Let $\alpha$ algebraic over $k$ of characteristic $p>0$ Prove that $\alpha$ is separable over $k$ if and only if $k(\alpha)=k(\alpha^p)$. Any suggestion, please.
6
votes
4answers
99 views

Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$

Is there any way to determine the Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$ not using the discriminant? Thanks!
0
votes
1answer
37 views

For a finite etale map $X\rightarrow T$ of degree $d$, and a $U$-point $t\in T(U)$, are there at most $d$ points in $X(U)$ lying over $t$?

If $f : X\rightarrow T$ is a finite etale morphism of connected schemes, and $U$ is another connected scheme and we're given a map $t : U\rightarrow T$, then must it be true that there are at most $d$ ...
1
vote
0answers
110 views

Any other mathematicians like Galois in recent history?

Any other mathematicians like Galois in recent history ? For "someone like Galois", I mean someone who developed a completely new theory all by himself, solved a big problem and the theory has big ...
1
vote
1answer
49 views

Proving degree $n$ have at last $n$ roots in $F_q[X]$

How to prove that in $F_{q}[X]$ of degree $n$, have $n$ roots?
0
votes
1answer
27 views

Prove that the Galois group of a polynomial $p(x)=x^q-1$ is the Cyclic group of order $q-1$, where $q$ is prime.

I understand why $q$ must be odd, since complex conjugation must be one of the Galios group's elements. But why must $q$ be prime?
4
votes
1answer
46 views

Finding a cubic polynomial whose splitting field over $\mathbb{Q}$ equals $\mathbb{Q}(a)$ if $a$ is any of its roots

Question: Let $\alpha$, $\beta$ and $\gamma$ be the roots of a rational cubic polynomial $q$. Can we find a (non-trivial) example where the splitting field of $q$ over $\mathbb{Q}$ equals ...
1
vote
1answer
60 views

What is $\hat{\mathbb{Z}}$?

I have been reading a bit about unramified extensions. If $K$ is $\mathbb{Q}$ or $\mathbb{Q}_p$ (p-adics), then there is a maximal unramified extension $K^{nr}$ of $K$. Then I have read in some notes ...
1
vote
0answers
28 views

determine the degree of an extension, check normality

I came across an old exam problem and I wonder if my solution is correct. Let $L=\mathbb{Q}(\omega)$, where $\omega=e^{\frac{2\pi i}{6}}$ is a primitive sixth root of unity: a) determine ...
3
votes
0answers
46 views

Solvability of Higher Degree Polynomials by Bring Radicals

A Bring radical of $a$ is any root of the polynomial $x^5+x+a$. It is known that we can solve the quintic if we're allowed to use the Bring radical. Now, I was wondering what happens if we ...
5
votes
2answers
122 views

Galois theory, had it solved any major problems beside its original applications to classical problems?

Galois theory, had it solved any major problems beside its original (classical) applications to roots of a fifth (or higher) degree polynomial equation (solvable algebraic equations and constructible ...
7
votes
2answers
84 views

Groups, inverse Galois problem and transcendence degrees

This is a curiosity of mine. I suspect there might be a trivial answer, but if there is none, this problem will probably haunt me for a long time... The question is as follows : Given a group $G$, ...
6
votes
1answer
95 views

Field extension $\mathbb Q(f)/\mathbb Q$ and its Galois group

Let $E/\mathbb{Q}$ and $F/E$ be finite extensions of fields, let $u$ be an element of $Aut(F/E)$, and let $f$ be an element of $F$. Suppose that (i) $[F:E]=3$, (ii) $F=\mathbb{Q}(f)$ and ...
2
votes
1answer
55 views

The Galois group of a polynomial

I was just looking for some clarification regarding the definition of the Galois group of a polynomial $f(x)$. So, if I remember correctly, this is defined as the Galois group of a splitting field of ...
0
votes
0answers
42 views

Why do nth roots (radicals) have closed forms whilst other polynomial roots do not?

The nth root function - $\sqrt[n]{a_{0}}$ - may be seen as an arithmetic operation (the inverse of the $pow(x, n)$ function) but it can also be interpreted as computing the roots of a specific class ...
4
votes
2answers
52 views

Non-isomorphic field extensions of $\mathbb{Q}$

I'm having a little bit of a problem with the following question: Show that there do not exist two irreducible polynomials $a(x)$ and $b(x)$ in $\mathbb{Q}[x]$ of degrees 6 and 7 respectively ...
2
votes
3answers
182 views

Showing Galois Group is Abelian

I'm having trouble showing that $\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ is Abelian. First I want to be able to show that $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is Galois, but I'm also not sure how to ...