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]
983 questions
3
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2 answers
Find the splitting Field of $x^4+x^2+1$
Find the splitting field of $$x^4+x^2+1=(x^2+x+1)(x^2-x+1)$$
I have $(-1±\sqrt{-3})/2$ and $(1±\sqrt{-3})/2$
so, $\mathbb{Q}(1,\sqrt{-3})$, but i do not make sure about that.
user146264
- 365
2
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2 answers
Splitting fields are not unique?
Let $F$ be a field and $0 \neq f \in F[X]$. I have proven that any two splitting field extensions $K_1,K_2$ are $F$-isomorphic.
Can anyone give an example of $2$ splitting field extensions of $f$ such that
$f$ has two different decompositions in…
user661541
2
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1 answer
Splitting field of a polynomial
I have $f(x)=x^7-6$ $\in \mathbb Q[x]$ I can see the roots are $e^{2\pi ik/7}\times6^{1/7}$ with $k$ from $0$ to $6$.
How can I show the splitting field $N$ has: $[N:\mathbb Q]=42$?
Daniel Moraes
- 789
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Find the splitting field of $x^6-2x^3-1$ over $\mathbb{Q}$.
Find the splitting field of $x^6-2x^3-1$ over $\mathbb{Q}$.
I know that the real roots of $x^6-2x^3-1$ are $\sqrt[3]{1+\sqrt{2}}$ and $-\sqrt[3]{-1+\sqrt{2}}$. And I know $\zeta_{6}$ is included somehow, but I'm honestly a little lost on how to…
user3566523
- 23
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1 answer
Is this definition of splitting field pleonastic?
I found this definition of a splitting field but I am wondering if the second condition does not implies the first one.
If $L$ is generated over $K$ by the zeros of the polynomials of the family, does not it follows that such zeros belong to $L$…
aleio1
- 995
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Finding degree of extension of splitting field of a polynomial
Let $K$ be the splitting field of $f(x)=x^{3}+πx+6$ over $F =\mathbb{Q}(π)$ and $K'$ be the splitting field of $g(x)=x^{3}+ex+6$ over $F'=\mathbb{Q}(e)$.
Is $[K:F]=[K':F']$ ?
$f(x)$ is irreducible over $\mathbb{Q}(π)$ and it has only one real root…
Naman
- 635
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1 answer
Splitting field construction
Context: Been teaching myself a little algebra - a lot of it makes more sense than in did years ago when I took that course.
Say $F$ is a field and $p,q\in F[x]$ are irreducible. We want to construct a splitting field for $pq$. I gather we do this:…
David C. Ullrich
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Degree of splitting field of $(x^{15}-1)(x^{12}-1)$ over $\mathbb{F}_7$.
I try to calculate the degree of splitting field of $(x^{15}-1)(x^{12}-1)$ over $\mathbb{F}_7$:
order of $7$ in $(\mathbb{Z}/15\mathbb{Z})^*$ is $8$;
order of $7$ in $(\mathbb{Z}/12\mathbb{Z})^*$ is $4$;
so degree is $\operatorname{lcm}(8,4)$; I'm…
user348628
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2 answers
Separability and normal closure
I am not sure about this problem.
Let $K/F$ be a finite separable extension and let $\widetilde{K}/F$ be a normal closure of $K/F$. Is $\widetilde{K}/F$ necessarily separable?
I tried considering $\alpha \in \widetilde{K} \backslash K$ and assume…
Koon
- 23
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1 answer
Prove that $Q(u)$ contains a subfield that has degree $3$ over $Q.$
Let $F$ be a splitting field of an irreducible polynomial over $\mathbb{Q}$, and $u$ be a root of the irreducible polynomial. Suppose that $[F:Q]=27$. Prove that $\mathbb{Q(u)}$ contains a subfield that has degree 3 over $\mathbb{Q}$.
user291864
- 41
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1 answer
Find the splitting field of the following polynomials.
Find the splitting field $E$ of the following polynomials and the degree of the extension
1) $X^4-1\in\mathbb Q[X]$
$X^4-1=(X-1)(X+1)(X^2+1)=(X+1)(X-1)(X+i)(X-i)$
therefore $E=\mathbb Q(i)\cong \mathbb Q[X]/(X^2+1)$
and thus $[E:\mathbb Q]=2$.
I…
idm
- 11,824
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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 roots of P(x)...
When is K different from G? IMO the splitting field is already the smallest field so there shouldn't be a…
user3698446
- 27
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1 answer
Just want to make it clear for the Defintion of Splitting Field
If $E = \mathbb{Q}(\alpha_1, \alpha_2, \alpha_3)$ the splitting field of $f(x)$ over $\mathbb{Q}$.
Then, does that have to be the case which $\alpha_1$, $\alpha_2$ and $\alpha_3$ are distinct roots of $f(x)$ ?
And also, does that have to be the case…
xxxxxx
- 603
0
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2 answers
Abstract algebra. splitting field
Find the degree of the splitting field of the polynomial $x^6-7$ over $F_{3}$ (the field with $3$ elements).
Here, $x^6-7$|${x^6}^2-7^2 = x^{12}-1$ where $7^2 = 1\mod 3$
After this step, I couldn't make anything. How can I continue for solution? or…
mathloverrr
- 23
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0 answers
Splitting field $\mathbb{Q}(\sqrt[11]{7},i)$
Let $K = \mathbb{Q}(\sqrt[11]{7},i)$. Is there a polynomial over $\mathbb{Q}$ whose splitting field is $K$?
I am not too sure of my answer. It is as follows:
It suffices to show that $K/\mathbb{Q}$ is not normal. Hence, we can conclude that $K$ is…
Noob4398
- 325