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).

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20 views

Looking for the splitting field of polynomials over $\mathbb{F}_4$

How to find the degrees of splitting fields of $x^7-1$ and $x^8-1$ over $\mathbb{F}_4$? For $x^8-1$ I wrote $$ x^8-1=(x^2+i)(x+ \sqrt{i})(x- \sqrt{i})(x+i)(x-i)(x+1)(x-1), $$ but I'm not sure what to ...
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47 views

What is the splitting field of $x^2 + 1$ over $\mathbb{F}_p$?

Is the solution always just the field itself?
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Finding splitting fields over $\mathbb{Q}(\sqrt{-3})$

What's the difference between finding a splitting field over $\mathbb{Q}$ and over $\mathbb{Q}(\sqrt{-3})$? Say, for example, we consider the polynomial $f=x^3-2$. Then over $\mathbb{C}$ this has ...
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Show that $Q(cos(40°))$ a normal extension of $Q$

I've calculated the minimal polynomial of $cos(40°)$ is $p(x) = 8x^3 - 6x + 1$. Since it has a degree of 3, the extension is finite. Also $(p', p) = 1$ so it has no multiple roots. I've tried writing ...
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Multiple choice question on extension of field

Let $\alpha=2^{1/5}$ and $\xi=e^{\frac{2\pi i}{5}}.$ Let $K=\mathbb{Q}(\alpha \xi) $ which of following are correct. There exist a field automorphism $\sigma$ of $\mathbb{C}$ such that $\sigma(K)=K$ ...
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1answer
23 views

Finite dimensional splitting field is generated by finite number of polynomials

Let $K$ be a field, and suppose $F$ is a splitting field over $K$ of a (possibly infinite) set of polynomials in $K[x]$. What I want to show is, that if $[F:K]$ is finite, then $F$ is a splitting ...
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Question for an abelian extension over $\mathbb{Q}$

Let $K\subseteq\mathbb{C}$ be the splitting field of the minimal polynomial of $\alpha+\beta$ over $\mathbb{Q}$, where both $\alpha$ and $\beta$ are algebraic real numbers, and suppose that the Galois ...
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On extension field

I have confusion on terminologies and results on extension field. If an extension field E is splitting field F as well as separable extension of F, Then (my doubt is), it is normal extension(from ...
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Find an example about splitting field

Let $f(x)$ be an irreducible polynomial in $\mathbb{Q}[x]$, and let $K$ be the splitting field of $f(x)$ over $\mathbb{Q}$. Now, suppose that $E$ is a splitting field of some polynomial in $\...
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Splitting field of $X^3-5$ and over $\Bbb{F}_p$, $p=7,11,13$ [closed]

Please help me to find the splitting field of $X^3-5$ over $\Bbb{F}_p$, $p=7,11,13$. Thanks,
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Why is the $\mathbb{Q}(i, \sqrt{3}, \sqrt[3]{5}) $splitting of the polynomial $(x^3 - 5)(x^2 + 1)$ and not just $(x^3 - 5)$?

I know that $\zeta = \sqrt[3]{5}$ satisfies the polynomial $x^3-5 \in \mathbb{Q}[x]$, and splits completely in $\mathbb{Q}(i, \alpha, \zeta)$, where $\alpha = \sqrt{3}$. I found the roots $x_1 = \zeta$...
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Does an algebraic closure of $F_p$ contain an element of infinite (multiplicative) order?

I am trying to find (as many as possible) elements in the algebraic closure of a positive characteristic field, being roots of irreducible polynomial inside some splitting field which are not roots of ...
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Question about finite extensions and splitting fields

Apparently any finite extension of a field with characteristic 0 is of the form $F(a)$ for some algebraic $a$. But wouldn't this mean that any two zeros of a polynomial (e. g. over $\mathbb{Q}$) would ...
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Non cyclic galois extension

Let $C$ be an algebraically close field of positive characteristic $p$ . I need to find a finite non cyclic galois extension of $C((T))$
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Splitting field in math software.

Can anyone help me to find a method for calculating the splitting field for a polynomial over a function field? I think this feature is not currently supported..
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Solving a polynomial congruence with rational number unknowns for absolute factorisation

I am implementing Gao's factorisation algorithm for bivariate rational polynomials $f\in\mathbb Q[x,y]$. An overview and the reference to the paper describing the algorithm are in this answer. I see ...
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1answer
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Splitting field of degree $p(p+1)$ contains a Galois subextension of degree $p$.

I've been studying for an algebra qualifying exam. Any help with the following result would be appreciated. Suppose $E$ is a splitting over $\mathbb{Q}$ of an irreducible polynomial $f(x)\in\mathbb{Q}...
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Finding Galois group of function field extension

Let $C((T))$ be the field of formal Laurent series in the variable $T$ over an algebraically closed field $C$ of characteristic $0$. I need to find to the Galois group of the splitting field for the ...
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1answer
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Splitting fields construction

My question is about the standrad construcion which shows that every polynomial has a splitting field. The construction is presented in wikipedia here. https://en.wikipedia.org/wiki/Splitting_field ...
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What do the elements of $\mathbb{F}_{243}$ look like?

Since $243 = 3^5$ and $3$ is a prime number we can construct the field $\mathbb{F}_{243}$ with $243$ elements, by taking an irreducible polynomial $f$ of degree $5$, then we get: $\mathbb{F}_{243}\...
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Is $\mathbb{Q}(\sqrt3 i) = \mathbb{Q}(\sqrt3, i)$?

Hi I am a little bit confused by extension fields and splitting fields. I have an exercise that asks me to find the degree of the extension $E/\mathbb{Q}$, where $E$ is the splitting field of the ...
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Splitting Field of $x^4 + x^3 + 1$ over $\mathbb{F}_{32}$

I'm trying to find the splitting field described in the title. I believe I have figured it out, but my method seems a bit involved and I'm wondering if there is any simpler way to obtain the result. ...
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How to show that the splitting field of $x^7 - 5$ over $\mathbb{Q}$ is equal to $\mathbb{Q}(\sqrt[7]{5}, \exp(2\pi i/7))$?

I have a polynomial $x^7 – 5$ over $\mathbb{Q}$. I know the splitting field $K = \mathbb{Q}(\alpha,\alpha\omega, …, \alpha\omega^6)$, where $\alpha = \sqrt[7]{5}$ and $\omega = \exp(2\pi i/7)$. Now, ...
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An irreducible polynomial in $\mathbb Q[x]$ that has all zeros with multiplicity $2$

I'm reading an Abstract Algebra textbook written by Jb-Fraleigh. When learning the Splitting field. I should first give the definition of a perfect field: Definition: Perfect field A field is ...
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Tower of galois extensions. Prove that $\mathbb Q(2^{1/3})$ is not contained in $K_n$

Prove that if $\mathbb Q=K_0 \subset K_1... \subset K_n$ is a sequence of fields such that $K_{i+1}/K_i$ are Galois extensions and $[K_{i+1}:K_i]=3 $ for each $i$, then $\mathbb Q(2^{1/3})$ is not ...
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Separability of the Normal Closure of a non finite extension

Gotta solve this problem: If $K|F$ is a separable field extension, must its normal closure $N|F$ be separable? Read another answer and when $K|F$ is finite, the answer is yes and the problem is ...
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Confusion about a statement in this theorem concerning normal field extensions

We were given a proof in class that is as follows: Given a finite extension $E/F$ the following are equivalent 1) $E/F$ is normal 2) If $\alpha \in E$ then the minimum polynomial of $\alpha$ over $...
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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 ...
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The fraction field of the ring of formal power series and a splitting field

Let the fraction field of the ring of formal power series over $\mathbb C$ be $\mathbb C((X))$. Let $D$ be the splitting field of $Y^n - X \in \mathbb C((X))[Y]$. Why is it also the splitting field of ...
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Show that $K$ is the splitting field of an irreducible polynomial of degree $n$ over F if $K/F$ is isomorphic to $S_n$ [duplicate]

Suppose that $K/F$ is Galois with Galois group $G(K/F) \simeq S_n$. Show that $K$ is the splitting field of an irreducible polynomial in $F[t]$ of degree $n$ over $F$. How can I go about showing this?
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Galois group of $x^6+1$ over $\mathbb{F}_2$

The question: Find the Galois group of $x^6+1$ over $F=\mathbb{F}_2=\{0,1\}$. EDIT: I have improved my attampt and would like for further help, thanks. My attampt: Let $E$ be the extension field ...
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1answer
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Is $[\mathbb Q(\sqrt[7]2):\mathbb Q]=7$ and $[\mathbb Q(w):\mathbb Q]=6$?

Let $[\mathbb Q(\sqrt[7]2):\mathbb Q]=a$ and $[\mathbb Q(w=e^{2\pi i/7}):\mathbb Q]=b$, then 1) $a=b$ 2) $a<b$ 3) $a>b$ $[\mathbb Q(\sqrt[7]2):\mathbb Q]=7$ as $x^7 - 2$ is an irreducible ...
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Zeros in Splitting Factor Rings

I am looking at the following proof that $x^3-x-1$ splits in an extension field of $F_{3}[x]$. Let us consider the field $$K = F_3[x]/\langle x^3-x-1\rangle$$ If $\theta$ is the image of $x$ in $K$, ...
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1answer
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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 \...
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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 ...
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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 $f$ ...
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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 ...
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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\...
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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, ...
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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)...
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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 ...
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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 ...
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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 ...
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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=\...
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1answer
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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 $...
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1answer
77 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)...
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154 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: $Y=-1\pm i \iff X=\sqrt[4]{2}e^{a\pi i/8}$, $...
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2answers
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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 ...
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1answer
35 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 ...
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13 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,...,\...