Articles of galois theory

The Galois group of two irreducible polynomials

What information is necessary to determine the Galois group of two irreducible polynomials? If I know the Galois group of $p(x)$ is $S_3$ and the Galois group of $f(x)$ is $S_2$ can I say the Galois group of $p(x)f(x)$ is $S_3 \times S_2$ ?

Cubic with repeated roots has a linear factor

If $f$ is a cubic polynomial with a repeated root over a field then $f$ has a linear factor. I think that is true in perfect fields but I don’t know how to prove it.

On the fields of rational fractions over $\mathbb{F}_{p}$

Let $K$ be a field with characteristic $p>0$. Let $K^{p}=\left\{ a^{p}:a\in K\right\}$. Prove that $K^p$ is a subfield of $K$. Furthermore, if $K=\mathbb{F}_{p}\left(X\right)$ is the fields of rational fractions over $\mathbb{F}_{p}$, find $K^p$ and $[K:K^p]$. Help me prove the second part. Thanks a lot.

Galois, normal and separable extensions

Theorem: Every finite extension, normal and separable is a Galois extension. Is the theorem equivalent to: $\mathbb K:\mathbb F$ is Galois $\iff \mathbb K:\mathbb F$ is normal & $\mathbb K:\mathbb F$ is separable ? thus, $\mathbb K:\mathbb F$ is not a Galois extension $\iff \mathbb K:\mathbb F$ is not normal or $\mathbb K:\mathbb F$ is […]

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 […]

Making the fundamental theorem of Galois theory explicit

I encountered the present question when investigating that other recent question of mine. Let $x_1,x_2, \ldots, x_8$ be indeterminates. Let $s_1,s_2, \ldots s_n$ denote the elementary symmetric polynomials (so that $s_1=\sum x_i, s_2=\sum_{i<j}x_ix_j$ etc. Let us consider also $$ \begin{align} g_1 &= x_1x_2+x_3x_4+x_5x_6+x_7x_8 \\ g_2 &= x_1x_3+x_1x_7+x_2x_4+x_2x_8+x_3x_5+x_5x_7+x_4x_6+x_6x_8 \\ g_3 &= x_1x_4+x_1x_8+x_2x_3+x_2x_7+x_4x_5+x_5x_8+x_3x_6+x_6x_7 \\ g_4 &= x_1x_5+x_2x_6+x_3x_7+x_4x_8 […]

Galois group over the field of rational functions

I am looking to find the Galois group of $x^3-x+t$ over $\mathbb{C}(t)$, the field of rational functions with complex coefficients. I have shown that the automorphisms of the rational function field $F(t)$ for fixed $F$ are precisely the fractional linear transformations that is $t \rightarrow \frac{at +b}{ct+d}$ for $a,b,c,d \in \mathbb{C}$. Is this useful? Also […]

Powers of $x$ as members of Galois Field and their representation as remainders

first question on math.stackexchange 🙂 I’m studying for a Cryptography – Communication Security exam, and it involves a certain quantity of number theory – finite field theory, so be warned: this is my first encounter with these topics, and you’ll have to be extra-clear with me 🙂 I thought I was doing pretty well with […]

A Finite Extension is Simple iff the Purely Inseparable Closure is Simple?

Question. Is it true that a finite extension $K:F$ is simple iff the purely inseprable closure is simple over $F$? I think have an argument to support the above. First we show the following: Lemma. Let $K:F$ be a finite extension and $S$ and $I$ be the separable and purely inseparable closures. Then $K=SI$. Proof. […]