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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Clarification about field extension and its degree

I know there are some posts about this, but I'm still confused regarding this specific question. It is said that the dimension of any field extension $\mathbb{Q}(w)$ is the degree of the irreducible ...
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Splitting field of $x^4+2$

While learning Galois theory, I tried to construct a splitting field for the polynomial $x^4+2$ over $\mathbb{Q}$, but I am terribly stuck. Since $x^4+2$ is irreducible by Eisenstein's criterion, I ...
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Extensions by Adjoining elements and Extensions by quotient of a Principal Ideal

Extensions can be constructed 2 ways to get an extension with roots of a polynomial Adjoining an element to a field - i.e. $F(\sqrt 2)$ is an extension of $F$. You can also build a tower of ...
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When is the splitting field of a polynomial different from the subfield containing all roots?

I've seen many times statements like: Let K be the splitting field of P(x) over F, let G be the subfield consisting of all the ...
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Proof checking - Galois theory

I am trying to show that if $K$ is a field and $f \in K[x]$ has exactly $n$ distinct roots, say $\alpha_1 ,..., \alpha_n \in L$ where $L$ is a splitting field (so $L=K(\alpha_1 , ..., \alpha_n )$) ...
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Splitting field of a polynomial over $\mathbb{Z}_p$

For $p$ prime, how can we show that the splitting field of $f(x)=x^{p-1}-1$ over $\mathbb{Z}_p$ coincides with $\mathbb{Z}_p$? The roots of $f(x)$ are the $p-1$ roots of unity. I think I am correct in ...
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Let $L$ be the splitting field of $x^5-2$ of $\mathbb{Q}$. Find every intermediate extension. [duplicate]

I'm looking at the example of how people did it with $x^4-2$ and trying to emulate that but it's still quite tricky. edit1: The splitting field will be $\mathbb{Q}(\sqrt[5]{2},\zeta_5)$ so one ...
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Galois group of splitting field of $x^4-16x^2+4$ isomorphic to $\mathbb{Z}/(2)\times \mathbb{Z}/(2)$

I've already shown irreducible. I know I need to get intermediate fields since they correspond to subgroups (which I can use for isomorphisms). But how do I get the intermediate fields here? edit: I'm ...
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Galois group of $x^3-2$ over $\mathbb{Q}(\omega)$

As titled. Let $\omega$ be a primitive cubic root of unity. I find the splitting field $E$ of $x^3-2$ over $F=$ $\mathbb{Q}(\omega)$ is $\mathbb{Q}(\omega,2^{1/3})$. Now I need to determine the ...
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Extension of an irreducible polynomial of degree $3$

Let $P = X^3 + X^2 - 2X -1$ be irreducible on $\mathbb{Q}$. After some calculations it is easy to show that the roots of $P$ are the real roots $r_k = 2 \cos \frac{2k\pi}{7}$ for $k=1,2,3$. Consider ...
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$X^4 +1$ Galois Group over $\mathbb{F}_5$

I had an exam question which was calculate the splitting field of $X^{40}-1$ over $F_5$ and then give its' Galois Group. Now I simplified the polynomial to products of polynomials of degree 1 to the ...
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Splitting field over a field and it degree. [duplicate]

I want to find the splitting field of $f(x)=x^4+2$ over the rationals, $\mathbb{Q}$; and the degree of that splitting field over $\mathbb{Q}$. So I first solved the equation $x^4=-2$, and get the ...
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Why does this polynomial splits over $\Bbb{F}_2(\sqrt{X})$

I have the following problem: Let $k=\Bbb{F}_2(X)$ and $E=\Bbb{F}_2(\sqrt{X})$. Then I know that the minimal polynomial of $\sqrt{X}$ over $k$ is $t^2-X$. But now in the lecture our prof. says that ...
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Splitting fields are normal using symmetric polynomials.

I was told it is possible to prove splitting fields are normal using the Fundamental Theorem of Symmetric Polynomials rather than the usual approach. Does anyone have hints or a reference for this? ...
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How do I prove the following fact about splitting fields?

Hello I have the following problem: Let $F$ be a splitting field of the polynomial $g(X)\in k[X]$. If $E$ is another splitting field of $g$ then there exists an isomorphism $\psi:E\rightarrow F$ ...
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Find the splitting field of $x^4-4x^2+1$ over $\mathbb{Q}$

I proved that the polynomial $x^4-4x^2+1$ is irreducible over $\mathbb{Q}$ , so if $a$ is a root of the polynomial, i got: $x^4-4x^2+1=(x-a)(x+a)(x^2+a^2-4)$. I don't know how to prove that the ...
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If the Galois group is $S_3$, can the extension be realized as the splitting field of a cubic?

We know that the Galois group of an irreducible cubic polynomial is $S_3$ or $A_3$, but is every group extension whose Galois group is $S_3$ a splitting field of a cubic polynomial? If not, the ...
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Splitting Field of $x^6 + x^3 + 2$ over $\mathbb{F}_3$

I need to decide if the splitting field of the polynomial $x^6 + x^3 + 2$ over $\mathbb{F}_3$ is isomorphic to $\mathbb{F}_{3^2}$. My argument is that $\mathbb{F}_{3^2}$ is splitting field of the ...
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How many degree $k$, monic polynomials factor completely in $\mathbb{Z}/p\mathbb{Z}$?

Suppose that we have the finite field $\mathbb{Z}/p\mathbb{Z}$, and we are considering adjoining the roots of degree $k$ monic polynomials to the field (whose coefficients are taken from this field). ...
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Question on why a field extension isn't a splitting field (and on general behavior of field extensions)

In class, we were discussing splitting fields, and I was wondering why the quotient of a polynomial ring by an irreducible polynomial "prioritizes" certain roots in the case that all of a ...
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Rupture field of an irreducible polynomial over a finite field equals its splitting field

Let $k= \mathbb{F}_{q}$ a finite field and $P \in k[X]$ an irreducible polynomial. Show that its rupture field is also its splitting field. My take : Let $K$ be the splitting field of P. By the ...
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If $[K:F]=p^{2}q$, where $p$ and $q$ are distinct primes, show that $K$ has subfields $L_{1}, L_{2}$ and $L_{3}$
Suppose that $K$ is the splitting field of some polynomial over a field $F$ of characteristic $0$.If $[K:F]=p^{2}q$, where $p$ and $q$ are distinct primes, show that $K$ has subfields $L_{1}, L_{2}$ ...