# Questions tagged [splitting-field]

The splitting field of a polynomial with coefficients in a field, $F$, is the smallest field extension to $F$ such that the polynomial decomposes into linear factors. Often used with (galois-theory).

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### Expressing the roots of a cubic as polynomials in one root

All roots of $8x^3-6x+1$ are real. (*) The discriminant of $8x^3-6x+1$ is $5184=72^2$ and so the splitting field of $8x^3-6x+1$ has degree $3$. Therefore, all three roots can be expressed as ...
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### Galois group of irreducible cubic equation

The polynomial $f(x) = x^3 -3x + 1$ is irreducible, and I'm trying to find the splitting field of the polynomial. We've been given the hint by our lecturer to allow an arbitrary $\alpha$ to be a ...
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### How to prove that algebraic numbers form a field?

I'd like to know how to prove algebraic numbers form a field, i.e., if $a,b$ are algebraic numbers, prove that $ab$ and $a+b$ are also algebraic numbers by finding specific polynomials that ...
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### Degree of a splitting field is no greater than $n!$

Is it possible to prove that if $P$ is a polynomial of degree $n$ then the degree of its splitting field is no greater than $n!$ without using notion of Galois group?
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### Proving $Gal(K/\mathbb{Q})$ is $D_4$. [duplicate]

Let $a = \sqrt{2+i}$ and $K$ is the splitting field of minimal polynomial of $a$ over $\mathbb{Q}$. Prove that $Gal(K/\mathbb{Q})$ is $D_4$. I find the minimal polynomial of $a$ is $p(x)=x^4-4x^2+5$ ...
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### Showing that $x^p - a$ either splits or is irreducible for characteristic $p$ (prime) in a field F.

Let $F$ be a field of characteristic $p$ and let $f (x) = x^p- a \in F[x]$. Show that $f (x)$ is irreducible over $F$ or $f (x)$ splits in $F$. If I assume that $f(x)$ doesn't split, then $f(x)$ ...
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### Prove that a polynomial with integer coefficients splits into linear factors modulo infinite number of primes

Let $f(x)$ be a polynomial over $\mathbb{Z}$. How to show that there is infinite number of primes $p$ such that $f(x)$ splits into linear factors over $\mathbb{F}_{p}$? Particularly, the task I was ...
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### Prove that $[\mathbb{Q}(\sqrt{4+\sqrt{5}},\sqrt{4-\sqrt{5}}):\mathbb{Q}] = 8$.

I'm trying to prove this result using elementary Field and Galois theory, but in an "efficient" way. It is desirable to avoid the use of powerful theorems of group theory or results about the ...
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### Galois group of $x^3+2x+2$

Let $f(x)=x^3+2x+2 \in \mathbb{Q} [x]$. Find the Galois group of $f$. I was trying to find the roots of $f$, but I couldn't. Teacher told us it wasn't necessary to find all roots. I know that since ...
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### How can we check that there exist such elements?

We have that $f=x^4-2\in \mathbb{Q}[x]$ and $E$ is the splitting field of $f/\mathbb{Q}$. It holds that $E=\mathbb{Q}[\rho, i]$, where $\rho^4=2, i^2=-1$. We have that $[E:\mathbb{Q}]=8$. According ...
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### $f(x)$ irreducible over $\mathbb Q$ of prime degree ; then Galois group of $f$ is solvable iff $L=\mathbb Q(a,b)$ for any two roots $a,b$ of $f$?

Let $f(x)\in \mathbb Q[x]$ be irreducible polynomial of prime degree , $L$ be its splitting field , then how to show that $f(x)$ is solvable by radicals over $\mathbb Q$ iff $L=\mathbb Q(a,b)$ for any ...
### Prove $x^n-a$ is irreducible over $\Bbb Q(\zeta_n)$
Here $\zeta_n$ denotes the primitive n-th root of unity. These days I am learning field theory. According to my lecture, for a radical extension we consider the splitting field of $x^n-a$ where $a$ ...