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]

876 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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### 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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### 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,…

TuoTuo

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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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### 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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### 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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### 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…

P.Styles

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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…

hermes

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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…

Hugo

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### 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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### 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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### Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group
with some elementary algebra,
$x - \sqrt{2 +\sqrt{2}} = 0 \implies x^2 = 2+\sqrt{2} \implies (x^2 - 2)^2 = 2…

oliverjones

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### 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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### Splitting field and Galois group of $(x^5-1)(x^2+1)$ over $\mathbb{Q}$

Consider $p(x) = (x^5-1)(x^2+1)$. Then, its splitting field is $\mathbb{Q}(e^{\frac{2\pi i}{5}}, i)$.
Thus, $f\in \text{Gal}(\mathbb{Q}(e^{\frac{2\pi i}{5}}, i)/\mathbb{Q})$ maps $\omega = e^{\frac{2\pi i}{5}}$ to any of $\omega^k$ for $k=1,...,4$…

idriskameni

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### Galois groups of $x^3-3x+1$ and $(x^3-2)(x^2+3)$ over $\mathbb{Q}$

I want to find the Galois groups of the following polynomials over $\mathbb{Q}$. The specific problems I am having is finding the roots of the first polynomial and dealing with a degree $6$ polynomial.
$X^3-3X+1$
Do we first need to find its…

thinker

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