Articles of splitting field

Find splitting field of a cubic polynomial

The problem is simple “Find the splitting field of $x^3+2x^2-5x+1$” Yeah because it’s too simple that I don’t know how explicit should the splitting field be. I mean we take 3 roots and let the desired field be the field generated by them. But can we find it more explicitly? Thank you

$f(x)$ irreducible over $\mathbb Q$ of prime degree ; then Galois group of $f$ is solvable iff $L=\mathbb Q(a,b)$ for any two roots $a,b$ of $f$?

Let $f(x)\in \mathbb Q[x]$ be irreducible polynomial of prime degree , $L$ be its splitting field , then how to show that $f(x)$ is solvable by radicals over $\mathbb Q$ iff $L=\mathbb Q(a,b)$ for any two roots $a,b$ of $f(x)$ ? i.e. how to show that $Gal(L/\mathbb Q)$ is solvable iff $L=\mathbb Q(a,b)$ for any […]

Dimension of a splitting field

Given a field $F$ and a polynomial $P \in F[x]$ such that $P$ is irreducible over $F$. Let $L_P$ be the splitting field of $P$ and $F$. Does $\operatorname{dim}_F({L_F}) = \deg(P)$ hold? If looking for a the minimal polynomial of $\alpha$ in a field $F$ is it sufficient to find a polynomial $P$ which is […]

Multiplicative relations and roots in different splitting fields

Let $f \in \mathbb{Z}[x]$ be a separable monic polynomial, with $f(0) \neq 0$, and $p$ be a prime number. Also, let $L$ be the splitting field of $f$ over $\mathbb{Q}_p$ and let $a_1, \ldots, a_n \in L$ be all the roots of $f$. Finally, let $b_1, \ldots, b_n \in \mathbb{C}$ also be the roots of […]

Finite field isomorphic to $\mathbb F_{p^n}$.

1) Let $p$ prime and $n\geq 1$ an integer. Show that there is a finite field of order $p^n$ in an algebraic closure $\mathbb F_p^{alg}$ and that all finite field is isomorphic to exactly one field $\mathbb F_{p^n}$ 2) Let $\mathbb F_q$ a finite field and $n\geq 1$ an integer. Let $\mathbb F_q^{alg}$ an algebraic […]

Showing that $x^p – a$ either splits or is irreducible for characteristic $p$ (prime) in a field F.

Let $F$ be a field of characteristic $p$ and let $f (x) = x^p- a \in F[x]$. Show that $f (x)$ is irreducible over $F$ or $f (x)$ splits in $F$. If I assume that $f(x)$ doesn’t split, then $f(x)$ cannot be written as a product of linear factors. But I can’t find any way […]

Find the splitting field of $x^3-1$ over $\mathbb{Q}$.

Find the splitting field of $x^3-1$ over $\mathbb{Q}$. My try: Factoring this to the most I can (in $\mathbb{Q}$), we get that $(x-1)(x^2+x+1)$ So $x=1$ is a root of $f(x)$. $x^2+x+1$ has no roots in $\mathbb{Q}$ so I conclude it is irreducible (since we have a low degree of $2$) Can I conclude that the […]

Galois group of $x^4-2$

I am trying to explicitly compute the Galois group of $x^4-2$ over $\mathbb{Q}$. I found that the resolvent polynomial is reducible and the order of the Galois group is $8$ using the splitting field $K=\mathbb{Q}(2^{1/4}, i)$. Hence I need to find 8 automorphisms. Thus do I just map elements of the same order to each […]

Galois Group of $x^{4}+7$

I am trying to find the Galois group of $x^{4}+7$ over $\mathbb{Q}$ and all of the intermediate extensions between $\mathbb{Q}$ and the splitting field of this polynomial. I have found that the splitting field is $\mathbb{Q}(i, \sqrt[4]{7}\sqrt{2})=\mathbb{Q}(i,\sqrt[4]{28})$. Edit: (Justifying the splitting field stated above) The roots of the polynomial are $\sqrt[4]{7}\big(\pm \frac{\sqrt{2}}{2}\pm i\frac{\sqrt{2}}{2}\big)$ so the […]

Is $\mathbb{Q}(\sqrt{3}, \sqrt{3})$ a Galois extension of $\mathbb{Q}$

From a previous question, we have that $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[4]{3})$, and I am assuming that we would need to use this to justify the answer to the question. Would it be right to use some idea of a splitting field here? Or is it something to do with $\mathbb{Q}(\sqrt[3]{3}, \sqrt[4]{3})$ not containing a primitive 3rd […]