Articles of splitting field

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 […]

How to prove that algebraic numbers form a field?

I’d like to know how to prove algebraic numbers form a field, i.e., if $a,b$ are algebraic numbers, prove that $ab$ and $a+b$ are also algebraic numbers by finding specific polynomials that satisfy $ab$ and $a+b$. I know there is a way to do this using Kronecker Symbol, but not sure exactly how to do […]

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. How to prove it?

Galois group of irreducible cubic equation

The polynomial $f(x) = x^3 -3x + 1$ is irreducible, and I’m trying to find the splitting field of the polynomial. We’ve been given the hint by our lecturer to allow an arbitrary $\alpha$ to be a root and then to check that $\alpha$ is a root, then $2-\alpha^2$ is also root. I’ve done this […]

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group

This question already has an answer here: Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$ 1 answer