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# 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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### Degree of $\mathbb{Q}(\sqrt[3]{2}, \sqrt{-3})$ over $\mathbb{Q}(\sqrt[3]{2})$

In Dummit and Foote, Abstract Algebra, Sec 13.4 Example 3, The splitting field of $x^{3}-2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt[3]{2}, \sqrt{-3})$. Then he says, quote : Since $\sqrt{-3}$ ...
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### Galois field splitting a polynomial

Can someone explain to me how i would go about doing a problem like this? I don't really know where to start. GF refers to a Galois field. I'm struggling to even understand exactly what they want me ...
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### same splittingfield of two polynomials $f(x)$ and $f(x+a)$

Show that for an element a in a field F f(x) and f(x+a) have the samsame splitting field. i want to get sure about my attempt : without loss of generality suppose deg f is at least 1 then in the ...
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### The splitting fields of two irreducible polynomials over $Z / p Z$ both of degree 2 are isomorphic

$p$ is a prime. Let $f_1, f_2 \in Z / p Z [t]$ both of degree 2 and irreducible. Show that they have isomorphic splitting fields. My approach was let $K_1 = F(\alpha_1, \beta_1) / F$ be the ...
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### Splitting field in multiple field extension

Let $F\subset K\subset E$ be fields, $f(x)\in F[x]$. E is the splitting field of $f(x)$ over $F$. Prove $E$ is the splitting field of $f(x)$ over $K$. My problem with this proof is that I do not know ...
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### Splitting field of a set of separable polynomials implies separability of extension.

Let $F$ be a splitting field of $S\subset K[x]$ over $K$, where $S$ is a set of separable polynomials. I want to show that $F$ is separable over $K$, meaning for all $u\in F-K$, the irreducible ...
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### Constructing splitting fields

Construct the splitting field for the polynomial $(x^4 - x^2 - 2)$ over $\mathbb Q$ (rationals) . What is the degree of the extension? Why? How would one go about tackling this question? I'm a bit ...
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I need to find the degree of the splitting field of $x^{4}-4$ over $\mathbb{Q}$ and $\mathbb{Z}/5 \mathbb{Z}$. I know that the roots are $\pm\sqrt{2}, \pm \sqrt{2}i$, so the splitting field is $F(\... 1answer 466 views ### Splitting field of x^3-2 as a simple extension Is there any elegant way to show that$\mathbb Q(\sqrt[3]{2}, w)=\mathbb Q(\sqrt[3]{2}+w)$, where$w=e^{i\frac{2\pi}{3}}$. I was thinking to show that$ 9+9 x+3 x^3+6 x^4+3 x^5+x^6$is the minimal ... 2answers 629 views ### Splitting Field of$x^6-6x^3+7$Given the polynomial$p(x)=x^6-6x^3+7 \in \mathbb Q$, find its splitting field$\mathbb F\subset \mathbb C$and the Galois group of the extension$\mathbb F /\mathbb Q$. In fact, the exercise asked to ... 2answers 1k views ### Splitting field of$x^n-a$contains all$n$roots of unity This statement is suggested as a correction to this question: If$K$is the splitting field of the polynomial$P(x)=x^n-a$over$\mathbb{Q}$, prove that$K$contains all the$n$th roots of unity. ... 0answers 107 views ### Splitting field containing$n$th root [duplicate] Let$K$be a splitting field of a polynomial over$\mathbb{Q}$. Suppose$K$contains an$n$th root of some number$a$. Then how can we show that$K$contains all the$n$th roots of unity? I don't ... 1answer 123 views ### Splitting field of irreducible polynomails Can I have two irreducible polynomials of different degree, having isomorphic splitting fields? The base field does not has to be perfect, . I mean if the base field is perfect, the extension is ... 1answer 441 views ### Splitting field of$X^n-a$Show that the splitting field of$X^n-a$over a field$K$is$K(\alpha, \zeta_n)$, where$\alpha$is a$n$-th root of$a$and$\zeta_n$is a primitive$n$-root of unity. 1answer 252 views ### Splitting field over$\mathbb{F}_3$The splitting field of$f(x)=x^8-1$over$\mathbb{F}_3$is$\mathbb{F}_{3^d}$where$d=ord_{(\mathbb{Z}/8\mathbb{Z})^*}(3)=2$. But$f(x)=(x^4+1)(x^4-1)$and$x^4+1$is irreducible over$\mathbb{F}_3$.... 1answer 158 views ### Splitting field of irreducible polynomial$f(x)=x^4 +ax^2+b$over field with charactersitic not equal to 2 this is from Seth Warner's Classical Modern Algebra: Let$K$be a field whose characteristic is not$2$, let$f=x^4+ax^2+b$be an irreducible polynomial over$K$, and let$L$be a splitting field of$...
I single variable polynomial splits completely in some field extension. Say $f(x,y,z)$ has a root $(a_0, b_0, c_0) \in \Bbb{Q}^3$. In a single variable we can say that if $a_0$ is a root then the ...