Mathematics

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).

1052 questions
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Galois group of $x^4-2$

I am trying to explicitly compute the Galois group of $x^4-2$ over $\mathbb{Q}$. I found that the resolvent polynomial is reducible and the order of the Galois group is $8$ using the splitting field $K=\mathbb{Q}(2^{1/4}, i)$. Hence I need to find 8…
Rutherford Mark
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the degree of a splitting field of a polynomial

Let $f(x)\in F[x]$ be a polynomial of degree $n$ . Let $K$ be a splitting field of $f(x)$ over $F$ . Prove that $[K:F] \mid n!$ . I only know that $[K:F] \le n!$ , but how can I show that $[K:F]$ divides $n!$ ?
Gobi
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Prove $f(x)=x^8-24 x^6+144 x^4-288 x^2+144$ is irreducible over $\mathbb{Q}$

How to prove $f(x)=x^8-24 x^6+144 x^4-288 x^2+144$ is irreducible over $\mathbb{Q}$? I tried Eisenstein criteria on $f(x+n)$ with $n$ ranging from $-10$ to $10$. None can be applied. I tried factoring over mod $p$ for primes up to $1223$. $f(x)$ is…
MaudPieTheRocktorate
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Galois Group of $x^{4}+7$

I am trying to find the Galois group of $x^{4}+7$ over $\mathbb{Q}$ and all of the intermediate extensions between $\mathbb{Q}$ and the splitting field of this polynomial. I have found that the splitting field is $\mathbb{Q}(i,…
Tuo
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Geometric interpretation of different types of field extensions?

In a first course on rings and fields we met the concept of field extensions, especially algebraic ones. The presentation of the material was very algebraic and felt a little lifeless. I was wondering whether there is some geometric way to think of…
Arrow
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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 polynomials in any one given root. Indeed, if $a$ is a…
lhf
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How to prove that algebraic numbers form a field?

I'd like to know how to prove that algebraic numbers form a field by using Kronecker Product, but not sure exactly how to do it. Edit: This question is different from the suggested duplicated one in that this question asks for an answer to prove…
John Lee
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Can I prove that a splitting field is normal without using zorn lemma

There is a theorem ： If $K \in F$ and $F$ is a splitting field of a polynomial in $K[x]$,then F is a normal extension over $K$. For proving this I choose a polynomial $g \in K[x]$ which has a root in $F$ and I want to prove that g splits completely…
Hugo
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Find the splitting field of $x^4+1$ over $\mathbb Q$.

Solution:Let $\mathbb E$ be the splitting field of $x^4+1$ over $\mathbb Q$ .Then $x^4+1$ splits into linear factors in $\mathbb E$ . $$x^4+1=(x^2-i)(x^2+i)=(x-\sqrt i)(x+\sqrt i)(x-\sqrt {-i})(x+\sqrt {-i})$$…
Styles
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When is a number in $\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?

Given an algebraic number $\alpha$ with minimal polynomial $P(x)$ of degree $2^n$, how can I decide if there are integers $a_1,\ldots,a_n$ such that $\alpha\in\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?
Charles
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Subfield of $\mathbb{Q}(\sqrt[n]{a})$

Exercise 14.7.4 from Dummit and Foote Let $K=\mathbb{Q}(\sqrt[n]{a})$ , where $a\in \mathbb{Q}$ , $a>0$ and suppose $[K:\mathbb{Q}]=n$ (i.e., $x^n-a$ is irreducible over $\mathbb Q$ ). Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d$ .…
user175968
11
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2 answers

splitting field of a polynomial over a finite field

I just realized that finding the splitting field of a polynomial over finite fields is not as "straightforward" as in $\mathbb{Q}$ I am struggling with the following problem: "Find the splitting field of $f(x)= x^{15}-2$ over $\mathbb{Z}_7=\Bbb…
Blood Borne
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Splitting field of an irreducible polynomial over $\mathbb{Q}$

Let $P(x)$ be an irreducible polynomial over $\mathbb{Q}$ ; I am interested its splitting field. I know that $\mathbb{Q}[x]/\langle P(x)\rangle $ have one of the roots of $P(x)$ , and that regardless of what root $\xi$ we take the field…
Belgi
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1 answer

Degree of splitting field less than n!

I've been asked to prove that if a polynomial $f\in \mathbb{Q}[x]$ has degree $n$, then the splitting field of $f$ has degree less than or equal to $n!$. That is, $[\mathbb{Q}(\alpha_1,...,\alpha_n):\mathbb Q]\leq n!$, where $\alpha_i$ are the roots…
Andrew Brick
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4 answers

Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$

I would like to know how to solve part $ii)$ of the following problem: Let $K /\mathbb{Q}$ be a splitting field for $f(X) =X^4-3X^2+5$ . i ) Prove that $f(X)$ is irreducible in $\mathbb{Q}[X]$ ii ) Prove that $K$ has degree $8$ over $\mathbb{Q}$ .…
Tom Oldfield
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