Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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).

0
votes
0answers
59 views

Why $\mathbb{Q}(\sqrt{2},\sqrt{5}):\mathbb{Q}$ is splitting field [on hold]

Why $\mathbb{Q}(\sqrt{2},\sqrt{5}):\mathbb{Q}$ is splitting field? Because is not second degree. What is another way to show it, in this example?
1
vote
1answer
50 views

Degree of splitting field of $f(x) \in \mathbb Q[x]$

I'm having trouble with a homework question, and I'm wondering if I can get some tips and/or hints (I do not want someone to solve the problem for me). The question is as follows: Let $f(x) \in \...
0
votes
0answers
26 views

Degree of splitting fields over finite fields

Suppose I have a finite field $F_{p^d}$, and I have a polynomial $f$ which is of degree $n$ and irreducible. I have a feeling that the splitting field of $f$ over $F_{p^d}$ is $F_{p^{dn}}$, but I am ...
0
votes
2answers
28 views

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 $f$ ...
0
votes
3answers
38 views

Degree of splitting field of $x^3-5$ over $\mathbb{F}_7$

Find the degree of the splitting field of $f(x):=x^3-5$ over $F:=\mathbb{F}_7$. Attempt: $f$ is irreduicible in $F[x]$ (suppose in contradiction it is reducible, thus it splits to at least one ...
4
votes
0answers
62 views

Prove that $[\mathbb{Q}(\sqrt[4]{10}\zeta_8,i):\mathbb{Q}]=8$

Determine the Galois group of $f=X^4+10$ and the lattice of intermediary fields. I know that $f$ is irreducible by Eisenstein for $p=5$. I calculated the roots to be $\sqrt[4]{10}\zeta_8^i$ for $i\in\...
0
votes
1answer
57 views

When degree of splitting field equals n factorial [duplicate]

Given that the degree of splitting field of a polynomial $f(x)$ over $\mathbb{Q}[x]$ is equal to $n!$ where $n$ is the degree of $f$, $n>2$. If $\alpha$ is a root of $f$ in the splitting field, ...
0
votes
0answers
42 views

For a field $K$ of characteristics $p>0$, when is a finite purely inseparable extension $F/K$ (with $[F:K]=p^n>1$) such that $F\cong K$?

An example that illustrates the question is: $F=\mathbb{F}_p(t)$ and $K=\mathbb{F}_p(t^p)$, for which $F\cong K$ by Luroth's theorem. Also, for $p>3$, consider $F=\overline{\mathbb{F}}_p(x,y)...
0
votes
0answers
25 views

Clarification on normality and separability in the context of field extensions. [duplicate]

I'm a little confused in regards to the definitions of normality and separability of field extensions in the context of Galois theory . The definitions seem very similar . In class they were defined ...
0
votes
0answers
28 views

Tower of Splitting Fields

If we have $F_n\geq\ldots\geq F_1$ fields such that $F_{i+1}$ is a separable splitting field over $F_i$ for all $i=1,\ldots,n-1$ is it true that $F_n$ is a splitting field over $F_1$? I think I can ...
0
votes
0answers
26 views

Splitting field of $x^p - 2$ over $\mathbb{Q}$ [duplicate]

Let $F = \mathbb{Q}$, $p$ a prime, and $f(x) = x^p - 2$. Let $K$ be the splitting field of $f(x)$ over $F$. Show that the Galois group $G = \operatorname{Gal}(K/F)$ is isomorphic to the multiplicative ...
6
votes
2answers
87 views

Prove that $[\mathbf{Q}(\sqrt{1+i},\sqrt{2}):\mathbf{Q}]=8$.

I am trying to calculate the Galois group of the polynomial $f=X^4-2X^2+2$. $f$ is Eisenstein with $p=2$, so irreducible over $\mathbf{Q}$. I calculated the zeros to be $\alpha_1=\sqrt{1+i},\alpha_2=\...
2
votes
1answer
102 views

Splitting field of $\sqrt{\vphantom{\sum}1+{\sqrt2}}$ and Galois group

Let $\alpha= \sqrt{\vphantom{\sum}1+{\sqrt2}}$. (a) Let $p(x)$ be the minimal polynomial of $\alpha$. Find $p(x)$. Let K be the splitting field of $p(x)$ (b)Let $E= \mathbb{Q}(i, \sqrt2)$ Show that $...
1
vote
1answer
66 views

If $f \in F[x]$ irreducible, ${\rm char}(F) = p$, then $f(x) = g(x^{p^e})$ and every root of $f$ has multiplicity $p^e$ in some splitting field

Let $f(x)$ be irreducible in $F[x]$, $F$ of characteristic $p>0$. Show that $f(x)$ can be written as $g(x^{p^e})$ where $g(x)$ is irreducible and separable. Use this to show that every root of $f(x)...
6
votes
3answers
148 views

Degree of splitting field of $X^4+2X^2+2$ over $\mathbf{Q}$

Find the degree of splitting field of $f=X^4+2X^2+2$ over $\mathbf{Q}$. By Eisenstein, $f$ is irreducible. By setting $Y=X^2$, we can solve for the roots: $Y=-1\pm i \iff X=\sqrt[4]{2}e^{a\pi i/8}$, $...
1
vote
2answers
44 views

Efficiently calculating the Galois group of $p(x)=x^4+4x^2-2$

I need to find the Galois group of $p(x)=x^4+4x^2-2$. Here is what I have done so far: By Eisenstein's criterion, $p$ is irreducible over $\Bbb Q$. Therefore, the Galois group is a transitive ...
0
votes
1answer
28 views

Show that $E/F$ is an abelian Galois extension such that $G=\text{Gal}(E/F)$ has exponent $m$ dividing $n$

Let $F$ be a field which contains $n$ distinct $n$th roots of $1$. Let $E$ be the splitting field over $F$ of a polynomial $$f(x) = (x^n−a_1)···(x^n−a_r)$$ with $a_i∈F$. Show that $E/F$ is an ...
0
votes
0answers
12 views

Tower relation for field degrees and separable polynomial in splitting field

I have the following example exercise: Let $K$ be a field and $L$ the splitting field of a separable polynomial $f\in K[X]$ of degree $n$. Denote the zeros of $f$ in $L$ by $\alpha_1,\alpha_2,...,\...
1
vote
1answer
72 views

Galois / Splitting field of $\ \sqrt{\vphantom{\sum}2+{\sqrt2}} $

Let $\xi$=$\ \sqrt{\vphantom{\sum}2+{\sqrt2}} $ (1) Find the minimal polynomial $r(x)$ $\in$ $\mathbb{Q}$ of $\xi$ (2)Prove that $\ \sqrt{\vphantom{\sum}2-{\sqrt2}} $ is also a root of $r(x)$ (3)...
0
votes
0answers
25 views

Relationship between $\Bbb Q(\sqrt2)$ and the splitting field of $x^2-2$

I've seen that $\Bbb Q(\sqrt2)=\{a+b\sqrt2:a,b\in\Bbb Q\}$ is the smallest field containing $\sqrt2$. Can this be realized if we consider the splitting field of $x^2-2$? $\frac{\Bbb Q[x]}{(x^2-2)}=\{...
2
votes
2answers
63 views

Splitting field/subfield is isomorphic

(1)Let $h$= $\mathbb{Q}$[t]/($t^2-2$). Show that there exists only one subfield of $\mathbb{R}$ isomorphic to $h$. (2)Let $h$= $\mathbb{Q}$[t]/($t^3-2$). Show that there exists three(3) subfields of ...
2
votes
2answers
71 views

Finding all polynomials $p(x)$ such that $ \mathbb{Q}[x]/p(x) \simeq \mathbb{Q}(A)$ for fixed $A$ algebraic

As an example, if we put $F =\mathbb{Q}(\sqrt{2}, \sqrt{3}, ... , \sqrt{n})$, $F$ is the splitting field of $p(x)$ so that we can write $$ p(x) = (x^2 - 2)(x^2 - 3)\cdots (x^2 - n). $$ Question: if ...
-1
votes
1answer
26 views

How to find the splitting field for $x^3-x^2-x-2$ over $\Bbb Q$? [closed]

I'm not quite sure on how to solve this one. Any help would be appreciated. :)
2
votes
1answer
55 views

If $f(x) \Bbb Q[x]$ has splitting field of degree $16$ over $\Bbb Q$ , are the roots of $f(x)$ solvable by radicals.

If $f(x) \Bbb Q[x]$ has splitting field of degree $16$ over $\Bbb Q$ , are the roots of $f(x)$ solvable by radicals. My attemt: Any general equation of degree $\geq 5$ is not solvable by radicals. ...
4
votes
1answer
84 views

Extension of Splitting Fields over An Arbitrary Field

Let $F$ be a field in which $0 \neq2$ in $F$, and consider $f=x^4+1$. If $E$ is the splitting field for $f$ over $F$, it turns out that $E$ is a simple extension of $F$. How does one realize this fact?...
0
votes
0answers
49 views

Show that for any element α of some extension of F, E(α) is a splitting field of f over F(α)

Let F be a field and let f ∈ F[X] be a polynomial with a splitting field E over F. Show that for any element α of some extension of F, E(α) is a splitting field of f over F(α) I'm not really sure how ...
0
votes
1answer
92 views

Did I find the splitting field of $x^3-3x+1$ over $\Bbb Q$?

I want to find the splitting field of $x^3-3x+1$ over $\Bbb Q$. I think I've got it but I'm not sure … Here's what I did : Using the formula for cubic roots I said that $$x=\dfrac{-b\pm\sqrt{b^2-...
0
votes
0answers
19 views

Is this how to find a polynomial with a given splitting field?

Suppose we want to find a polynomial whose splitting field is $\Bbb Q(\sqrt{2}, \sqrt{-3})$ over $\Bbb Q$. Then the following is how you'd do it right ? We want to adjoin the roots $\alpha=\sqrt{2}, \...
3
votes
1answer
90 views

Let $f(x)=x^4-x^2-1 \in \Bbb Q[x]$, $K$ is the splitting field of $f$ over $\Bbb Q$.

Let $f(x)=x^4-x^2-1 \in \Bbb Q[x]$, $K$ is the splitting field of $f$ over $\Bbb Q$. I’m to find out $\text{Gal}(K/\Bbb Q)$, how do I do? If the four roots of $f(x)$ in $\Bbb C$ are $x_1, x_2, x_3, ...
2
votes
0answers
33 views

automorphism group of splitting field of cubic polynomial over $\mathbb{Q}$

The question is to compute the isomorphism class of automorphism group of the splitting field over $\mathbb{Q}$ of $x^3 - 6x + 2$. I know that this polynomial is irreducible over $\mathbb{Q}$ by ...
1
vote
1answer
42 views

Dimension of splitting field as the order of an element mod n

Let $p$ be a prime number, $n\in\mathbb{N}$, with $p \nmid n$. Let $K$ be the splitting field of $x^n-1$ over $\mathbb{F}_p$. Show that the dimension of $K$, $[K:\mathbb{F}_p] = d$, is the smallest ...
2
votes
1answer
50 views

Degree of splitting field of $X^n-1$ over some finite field

Let $k$ be a finite field of order $q$ in characteristic $p$, let $n$ be a positive integer not divisible by $p$, and let $K$ be the splitting field of $X^n-1$ over $k$. Prove that $[K:k]$ equals the ...
0
votes
1answer
27 views

Algebraic closure not even finitely generated over base field [closed]

Show the algebraic closure $\bar{\mathbb{Q}}$ is not even finitely generated over the field $\mathbb{Q}$ I'm not sure how to go about this..
3
votes
1answer
69 views

$f(x)$ is irreducible over finite field $F_p$ , does $f(x)$ split over $F_p / \langle f(x) \rangle$?

I want to know whether it is possible to generalize the next thing exercise. $x^4 + x + 1$ is irreducible over $F_2[x]$ , so $F_2[x]/\langle x^4 +x+1\rangle = span_{F_2} \{1,\bar{x} , \bar{x}^2,\bar{...
2
votes
1answer
87 views

Splitting field of separable polynomial is Galois extension

Definitions: $f$ is separable if every irreducible factor has distinct roots. $E/F$ is a Galois extension if the fixed field of the Galois group Gal$(E/F)$ is $F$ I would like to prove the following ...
1
vote
1answer
66 views

Can splitting field be generated by one root?

Say $f$ is an irreducible polynomial over a field $F$, and $\alpha$ is one of its roots, then is $F(\alpha)$ a splitting field for $f$? I tried to find some counterexample, but I failed.
2
votes
2answers
45 views

Is $[\Bbb Q(5^{1/2}, 5^{1/7}): \Bbb Q(5^{1/7})]$ a normal extension?

I've working on a problem set with a bunch of these, and I get the idea generally. An extension is normal if all of the roots for the min pol for the element we are extending by are in the field. e.g.,...
0
votes
0answers
14 views

prove that $K$ is the composite field of $K_i $'s

Let $\{f_i\}_{i \in I}$ be a family of polynomials in $F[X]$ (F is of course some field). Consider the extension field $K$ such that $f_i$ splits in $K[X]$ and is generated by all the roots of $f_i,i\...
2
votes
1answer
68 views

Degree of splitting field for $f(x) = x^4 - x^2 + 4$ over $\mathbb{Q}$

I started for finding the roots of the polynomial (4 in total) which took the forms $$ \pm \sqrt{\frac{1}{2}\left( 1 \pm i \sqrt{15} \ \right)} $$ I figured that adjoining the positive square root ...
0
votes
1answer
27 views

Splitting Fields over arbitrary fields

I am currently learning field theory by myself (following Advanced Modern Algebra by Joseph Rotman at Chapter 3). We suppose $\mathbb{F}$ is some arbitrary field and $P\in\mathbb{F}[x]$, the ...
2
votes
1answer
121 views

(Degree of) Splitting Field for $f(x) = x^p - 2$ over $\mathbb{Q}$, p prime

Here is the work I've done so far: $\sqrt[p]{2}$ is a real root of $f(x)$ Any $(\zeta \sqrt[p]{2})$ where $\zeta$ is a $p^{th}$ root of unity is also a root of $f(x)$ Since $p$ is prime, all its ...
2
votes
1answer
46 views

Determine $[\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11}):\mathbb{Q}]$, and determine if $\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11})/\mathbb{Q}$ is normal

Determine $[\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11}):\mathbb{Q}]$, and determine if $\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11})/\mathbb{Q}$ is normal. If we let $x = \sqrt[4]{11}+i\sqrt[4]{11} = \sqrt[4]{...
1
vote
1answer
43 views

Find the splitting field $K$ of $x^{12}-9$ over $\mathbb{Q}$ and determine $[K:\mathbb{Q}]$.

Find the splitting field $K$ of $x^{12}-9$ over $\mathbb{Q}$ and determine $[K:\mathbb{Q}]$. My approach: First we can factor it $x^{12}-9 = (x^6-3)(x^6+3)$ so that the first factor gives us that $\...
2
votes
1answer
31 views

Splitting field of $f=t^{4}+2\in \mathbb{Z}_{3}[t]$

In order to determine the splitting field of $t^{4}+2\in \mathbb{Z}_{3}[t]$, I first "guessed" the roots $1$ and $2$ then, by polynomial division, obtained $t^{4}+2=(t^{2}+1)(t+2)(t+1)$ Since for ...
0
votes
1answer
26 views

The degree of a splitting field over $\mathbb{F}_p$ of non-reducible monic polynomial of degree $n$

Let $f\in\mathbb{F}_p(x)$ be a monic irreducible polynomial, denoting $\deg(f)=n$. I wish to show (if it's true) that $f(x)$'s splitting field is $\mathbb{F}_{p^n}$. I did some manual test for some ...
2
votes
2answers
104 views

Prove that $[ \mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]=8.$

I have to solve the following exercise: Compute $[\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]$ and $\operatorname{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})/\mathbb{Q}).$ Here my attempt: ...
0
votes
0answers
47 views

Splitting field of an irreductible polynomial $f(X) \in F_{q}[X]$

Let $F_q$ be a finite field ($q$ is a power a prime) and irreductible polynomial $f(X)\in F_q[X]$ with degree $n\geq 2$. I have to see that $F_{q^n}$ is the splitting field of $f$ over $F_q$, and ...
5
votes
2answers
176 views

Prove that $[\mathbb{Q}(\sqrt{4+\sqrt{5}},\sqrt{4-\sqrt{5}}):\mathbb{Q}] = 8$.

I'm trying to prove this result using elementary Field and Galois theory, but in an "efficient" way. It is desirable to avoid the use of powerful theorems of group theory or results about the ...
1
vote
0answers
31 views

Separable polynomial and characteristic of field

Suppose $f(x)$ is a monic separable polynomial over $F$ such that the set of all roots of $f(x)$ in the algebraic closure $\bar{F}$ is a subfield of $\bar{F}$. Then, the splitting field of f(x) over ...
0
votes
2answers
38 views

Complex conjugate and Field Extension

Let $E$ be a subfield of $\mathbb{C}$ and Let $\overline{E}=\{\overline{z} \, |\, z \in E \}$ with $\overline{z}$ being the complex conjugate of $z$. Let $K$ be a subfield of $\mathbb{C}$ with $\...