# 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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### Calculating the splitting field of this polynomial

I was wondering if anyone could check to see if i have done the following problem correct. I want to find the splitting Field for $f(x)$ over $\mathbb Q$, $E$ say, and also evaluate $[E:\mathbb Q]$ ...
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### Prove that no finite field is algebraically closed (using a particular method)

I'll preface this question by saying this question is not a duplicate as I am looking for insight into proving this a particular way. I have found a proof of the result both here on stackexchange and ...
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### Calculating the splitting field of a given polynomial.

I was wondering if anyone could explain how i can find the splitting field for $f(x)$ over $\mathbb{Q}$, $E$ say, and also evaluate $[E:\mathbb{Q}]$ for a basic case of $f(x)=x^{4}-15$ I know that i ...
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### How to find subgroups, fixed fields and subfields of Galois group?

Let $\Sigma$ be a splitting field in $\mathbb C$ of the polynomial $f(x)=x^4-49$ over $\mathbb Q$. Construct $\Sigma$ and find the degree of the field extension $\Sigma:\mathbb Q$. Determine the ...
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### What do you mean by splitting field? Find the splitting field of $x^5 -1$

Okay, first of all sorry for a stupid question. Now what i know is - Splitting Field: Let K be a field and let f(x) = a0 + a1 x¹ + a2 x² · · · + an xⁿ be a polynomial in K[x] of degree n>0. An ...
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### Algebraic closure of a real-closed field

There is a theorem that says: $K$ is a real-closed field if and only if (1) $x^2+1$ is irreducible in $K$ and (2) $K(i)$ is algebraically closed (where $i$ is a root of $x^2+1$ in an ...
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### Splitting field of $x^2+[1]$ over $\mathbb Z_2$

While finding splitting field of $x^2+[1]$ over $\mathbb Z_2$ I found that all roots of $x^2+[1]$ lies in $\mathbb Z_2$ than how can I find splitting field ?
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### A question on the definition of splitting field

The definition of splitting field is as follows: Suppose that $L$ is a field extension of $K$, and that $f ∈ K[X]$. We say that $f$ splits completely over $L$ if there exist $c,α_1,α_2,...,α_n ∈ L$ ...
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### Find the splitting field of a polynomial over $Z_{p}$

this question is from the book Graduate Algebra written by Lang. It asks me to find the splitting field of $f(x)=x^{p^{8}} -1$ over the field $Z_{p}$. I got a proof from my friend but I don't really ...
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### Set-theoretic equality of splitting fields within a fixed algebraic closure

Let $F$ be a field and let $f(x)\in F[x]$ be a polynomial. Recall the following two facts: (1) algebraic closures are unique up to isomorphism (2) splitting fields are unique up to isomorphism Fix ...