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).
Questions tagged [splitting-field]
1044 questions
27
votes
2 answers
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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23
votes
1 answer
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
- 7,704
20
votes
5 answers
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 $1223$. $f(x)$…
MaudPieTheRocktorate
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19
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1 answer
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
- 4,882
15
votes
1 answer
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
- 14,390
14
votes
1 answer
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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14
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4 answers
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…
Eugene Zhang
- 17,100
13
votes
1 answer
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…
Hugo
- 606
13
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1 answer
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…
Styles
- 3,609
12
votes
1 answer
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
- 32,999
11
votes
3 answers
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
- 23,614
11
votes
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…
Andrew Brick
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10
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2 answers
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 $[E:\mathbb{Q}]=d$. Prove…
user175968
10
votes
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 $\mathbb{Q}$.
iii)…
Tom Oldfield
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10
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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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