# Tagged Questions

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 (galois-...

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### Galois group of $x^{2^k}+1$

What is the Galois group of $f(x)=x^{2^k}+1$ over $\mathbb{Q}$?
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### monomorphism on an algebraic field extension

let $E|K$ be an algebraic extension and $\phi:E\rightarrow E$ a $K$-algebra monomorphism,prove that $\phi$ is onto. i assume $\alpha\in E-\phi(E)$ to make a contradiction and i assume $f(x)$ to be ...
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### Compute the degree of the splitting field

I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial ...
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### product considering the period of index over a cyclotomic extension

This is an exercise from Milne Galois Theory (Chapter 3 Exercise 13). Let p be an odd prime, and let $\zeta$ be a primitive $p^{\text{th}}$ root of 1 in $\mathbb C$. Let $E = \mathbb Q[\zeta ]$, and ...
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### Prove that this extension is Galois with galois group the quaternions

Let $M=\mathbb Q(\sqrt{2},\sqrt{3})$ and let $E=M(\sqrt{(\sqrt{2}+2)(\sqrt{3}+3)})$. Prove that: M is Galois over $\mathbb{Q}$ Show that $E$ is Galois over $\mathbb Q$ with Galois group the ...
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### galois group of a biquadratic involving primes.

Let $f(x)=x^4-px^2+q \in \mathbb Q[x]$ be a polynomial with $p,q$ be distinct primes. Prove that $f$ it's irreducible over $\mathbb Q$. Prove that it's Galois group is the dihedral. I proved the ...
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### Example of a non-separable normal extension

I'm trying to give an example of a normal field extension $K|F$ that is not separable. I now that if $F$ is finite or char$(F)=0$, $K|F$ is automatically separable, thus, I must look into infinite ...
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### Calculate the degree of the field extensions.

I have been staring at this question for a while. I'm sure there is a little trick I am missing...anyway, it is the following: $f = x^3 + x + 3$ a) Show $f$ is irreducible over $\mathbb{Q}[x]$: ...
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### Intermediate fields of Gal($x^5-2$, $\mathbb{Q}$)

I'm having trouble findind the fixed field of each subgroup of the Galois group of $\mathbb{Q(\alpha, \omega)}$, where $\alpha = 2^{\frac{1}{5}}$ and $\omega$ is a 5-th primitive root of unity. I list ...
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### Some Galois theory

I have a question on field extensions, and I can't seem to find precise answers when browsing through online notes etc. Here it is: suppose $K$ and $k$ are fields with $k \leq K$ and $[K : k] = m$ ...
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### 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 ...
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### 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 work ...
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### 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
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### 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 ?
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### 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 $3$ ...
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### 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 $L$. ...
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### 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 $K$ ...
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### 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 ...
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### $[F(t):F(t^n)]=n$ 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 ...
### 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 ...
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 \$[K_{i}:K_{i-1}]...