# 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).

571 questions
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### Splitting field of $f=t^{4}+2\in \mathbb{Z}_{3}[t]$

In order to determine the splitting field of $t^{4}+2\in \mathbb{Z}_{3}[t]$, I first "guessed" the roots $1$ and $2$ then, by polynomial division, obtained $t^{4}+2=(t^{2}+1)(t+2)(t+1)$ Since for ...
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### The degree of a splitting field over $\mathbb{F}_p$ of non-reducible monic polynomial of degree $n$

Let $f\in\mathbb{F}_p(x)$ be a monic irreducible polynomial, denoting $\deg(f)=n$. I wish to show (if it's true) that $f(x)$'s splitting field is $\mathbb{F}_{p^n}$. I did some manual test for some ...
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### Prove that $[ \mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]=8.$

I have to solve the following exercise: Compute $[\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]$ and $\operatorname{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})/\mathbb{Q}).$ Here my attempt: ...
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### Splitting field of an irreductible polynomial $f(X) \in F_{q}[X]$

Let $F_q$ be a finite field ($q$ is a power a prime) and irreductible polynomial $f(X)\in F_q[X]$ with degree $n\geq 2$. I have to see that $F_{q^n}$ is the splitting field of $f$ over $F_q$, and ...
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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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### Separable polynomial and characteristic of field

Suppose $f(x)$ is a monic separable polynomial over $F$ such that the set of all roots of $f(x)$ in the algebraic closure $\bar{F}$ is a subfield of $\bar{F}$. Then, the splitting field of f(x) over ...
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### Is the following proof regarding the degree of a splitting field correct?

Some hours ago I was going through this post and I thought about the following argument to prove the result. For the sake of completeness I will be mentioning both the result and my attempt to prove ...
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### When is $x^{2^n} + 1$ Reducible in $\mathbf{F}_p$ For All Primes $p$

Define $f_n = x^{2^n} + 1$. Then we want to show that there is an integer $n$ such that $f_n$ is reducible in $\mathbb{F}_p[x]$ for all primes, $p$. However, I want to do this using the hint The ...
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### Showing Two Polynomials Have Isomorphic Splitting Fields over $\mathbb{F}_p$

Consider the polynomials, $f_1, f_2 \in \mathbb{F}_p[x]$, given by \begin{align} f_1 &= x^{2^n} + 1 \\ f_2 &= x^{2^{n+1}} - 1 \end{align} How can we show that $f_1$ and $f_2$ have ...
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### Existence of polynomials in $\mathbb{Q}[x]$ and $\mathbb{Z}[x]$ with same splitting fields.

I've been asked to prove the following statement in a Galois Theory Seminar after being introduced to Dedekind's Theorem. (I assume this could potentially help getting the answer.) Let $f(x)$ be a ...
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### Show that $R$ is a field.

let $E$ be a splitting field of a polynomial over $k$. Let $R$ be a ring such that $k\subset R \subset E$ Then $R$ is a field. Does not it follow from the fact that any integral domain which is ...
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### degree of splitting field of palindromic polynomial

Let $p(x) \in \mathbb Q[x]$ be a palindromic polynomial of even degree 2n. Let $K$ be the splitting field of $p(x)$. Prove that $[K:\mathbb Q] \leq 2^nn!$. I know that the palindromic polynomial ...
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### Question on splitting fields.

Let $K$ be a field, $f(x)\in K[x]$ monic and irreducible polynomial over $K[x]$, $E$ the splitting field of $f(x)$ over $K$ and $a,b\in E\$ two roots of $f(x)$. We would like to prove that there ...
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### Find Galois group of $f(X) = X^5+4X^3+9X+3$ over $\mathbb{Q}$, and find $K_{2}$, the splitting field of $f\pmod 2$

Here's the full problem that I'm trying to solve, and my partial progress: Let $f(X) = X^5+4X^3+9X+3 \in \mathbb{Q}[X]$, and let $f_{2}(X) \in \mathbb{F}_{2}[X]$ and $f_{3}(X) \in \mathbb{F}_{3}[X]$...
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### Expression of elements in splitting field

I was reading $16.3$ SPLITTING FIELD in Algebra by Artin, where I met the following: For every element $\beta$ of $K$, there is a polynomial $p(u_1,\cdots, u_n)$ with coefficients in $F$, ...
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### Showing that the Galois group has an element of order 8.

Let $K = \mathbb{R}(X)$ be the field of rational functions with real coefficients, and let $$F = \mathbb{R}\left(X^4 - \frac{1}{X^4}\right)$$ be a subfield of $K.$ Let $L$ be the Galois closure of the ...
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### Galois number fields that have the imaginary unit.

$\newcommand\Q{\mathbb Q}$Is it possible to find an irreducible polynomial $f\in \Q[x]$ of degree $4$ such that the following holds: All roots of $f$ are non-real The splitting field of $f$, $K_f$, ...
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