# 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).

949 questions
Filter by
Sorted by
Tagged with
1 vote
52 views

### Splitting field and degree of extension of $x^4 + 2x^2 -2$ over $\mathbb F_{13}$

Splitting field and degree of extension of $x^4 + 2x^2 -2$ over $\mathbb Q$ and $\mathbb F_{13}$ I already solved it over $\mathbb Q$, but I don't know how it's done over finite fields. I tried ...
43 views

### How do I prove that the primitive element of a field extension are this way.

I'm doing an introductory course of field theory and there is one excercise that as easy as it seems it bring me on my nerves. It states: Let $α_1, \dots, α_n ∈ \mathbb{C}$ be the roots of an ...
22 views

### linearly disjoint field extensions in terms of algebraic extension

I have seen the following definition for the linearly disjoint field extensions $E$ and $F$: "Two extension fields $E$ and $F$ of a field $k$ contained in a common field $L$, such that any finite ...
63 views

42 views

### Intermediate fields of a Galois extension

Let $K$ be the splitting field of $f(X)=X^3-2$ over $\mathbb{Q}$. I am asked to find complete list of intermediate fields $k$, $\mathbb{Q}\subseteq k\subseteq K$ such that $[k:\mathbb{Q}]=3$. I've ...
80 views

1 vote
74 views

### On the definition of algebraic closure

Let $F$ be a field. By definition, the following are equivalent: $F$ is algebraically closed. Every nonconstant polynomial in $F[x]$ splits over $F$. Every nonconstant polynomial in $F[x]$ has a ...
39 views

1 vote
39 views

### Relation between number of roots of a polynomial and number of cosets of some subgroups of Galois group

Suppose $\mathbb{F}/\mathbb{K}$ is a field extension, and $f(x) \in \mathbb{K[x]}$ is minimal polynomial of $u \in \mathbb{F}$. Let $G=Aut_{\mathbb{K}}\mathbb{F}$ be the group of field automorphisms ...
120 views

132 views

### How to find a $\mathbb{Q}$-basis for the splitting field of $x^4-2$ over $\mathbb{Q}$?

Let $\mathbb{K}$ be the splitting field of $x^{4}-2$ over $\mathbb{Q}$ with $\mathbb{K}\leq \mathbb{C}$ a field extension. Determine a $\mathbb{Q}$-basis for $\mathbb{K}$. First, denote $f(x)=x^{4}-2$,...
1 vote
47 views

61 views

### Galois Group of $\mathbb{Q}(\sqrt 2, \sqrt 3)$

Im trying to compute the Galois group of the polynomial $(x^2-2)(x^2-3)$ which has as a splitting field $\mathbb{Q}(\sqrt 2, \sqrt 3)$. The extension is Galois with degree $4$ hence the group has $4$ ...
1 vote
230 views

55 views

### Splitting Field of $p(x) = (x^4-1)(x^3-2)$ [closed]

I found the splitting field of a polynomial $p(x)=(x^4-1)(x^3-2)$ to be $\mathbb{Q}(\sqrt{2},\omega,i)$ where $\omega=-\frac{1}{2}+\frac{\sqrt{3}}{2}i$, but is it possible to simplify this field?
55 views

### Find all homomorphisms from $\mathbb{Q}(\sqrt2)$ to $\mathbb{Q}(\sqrt2, \omega)$

We need to determine all $\mathbb{Q}$-homomorphisms from $\mathbb{Q}(\sqrt2)$ to $\mathbb{Q}(\sqrt2, \omega)$, where $\omega=e^{2\pi i/5}$. I can figure it out that there are 5 injective ...
19 views

99 views

20 views

### Galois group of a given splitting field.

I’m studying Galois theory, and have made my mind into seeing that there is a strong conection between a polynomial and it’s Galois group, but i’ve also seen that the Galois group is isomorphic to the ...
111 views

51 views

### How do I find this polynomials splitting field over $\mathbb{Q}$?

I’m trying to find the splitting field of the polynomial p(x) = $x^8 + 11x^4 + 24$. Here is my attempt: The obvious solutions to the polynomial are $x = \pm \sqrt{-3}$, and $x = \pm \sqrt{-8}$. ...
59 views

### Correspondent subfield of a subgroup of the Galois group

I'm trying to find all the subfields of the splitting field of $x^4-2$. It's $\mathbb{Q}(\sqrt{2}, i)$. I already figured out the Galois group is isomorphic to $D_4$ and I found the correspondent ...
91 views

### Show: No polynomials in $\mathbb{Q}[X]$ satisfy $f(X)^3-f(X)+2=(X^4-7)*g(X)$

Prove: There exist no polynomials $f,g$ in $\mathbb{Q}[X]$ satisfying $$f(X)^3-f(X)+2=(X^4-7)*g(X)$$ As a hint I got: Consider the $4th$ root of $7$. So for $X=7^{1/4}$: $f(7^{1/4})^3-f(7^{1/4})+2=0$. ...
1 vote
Let $F$ be a field with an $n$-th root of unity. Let $a,b ∈ F$ such that $f(x) = x^n - a$ and $g(x) = x^n - b$ are irreducible. Show that $f$ and $g$ have the same splitting field, iff $b=c^na^r$ for ...