Let $f$ be a polynomial of degree $n$ over the finite field $\mathbb{F}_p$. We aim to find its Galois group. If $f$ is irreducible and separable, then $\mathbb{F}_{p^n}$ is its splitting field, and its Galois group is $Z/nZ$, which is generated by $\sigma:x\mapsto x^p$. Now, suppose we do not...

I want to find the Galois group $G$ of $f=x^5+2$ over $\mathbb{Q}$. I will write my approach, and I would like to know if there is a faster approach or a more standard one that does not require creativity. Let $\alpha$ be such that $\alpha^5=-2$ and $\zeta$ be the fifth root of unity. Then, the...

An exercise in Lang asks us to find the Galois group of $$f=x^6+22x^5-9x^4+12x^3-37x^2-29x-15$$ over the rationals. I am going to write as far as I can. Then, I will ask how to proceed. I also welcome any other suggestions to solve this problem. Step 1: Reducing mod 2, we get...

I have zero, one, addition, subtraction, multiplication, division (by a non-zero expressible number), n-th roots (of positive expressible numbers, where n is expressible), and all the trigonometric functions (in their default domain restricted to the expressible numbers). The number expressible...

I am a high school student who, after some tinkering, came across the Abel–Ruffini theorem. I then learned about its explanation through Galois theory, particularly the result that a polynomial is solvable by radicals if and only if its Galois group is solvable. This leads to a confusion. The...

I realize this may be an unusual request, but I am trying to find out whether this style of studying mathematics already exists, or whether there are resources that come close to it. I am not looking for a standard textbook on Galois theory, nor for a historical overview followed by the modern...

It is a classical result that the roots of a general cubic polynomial $x^3 + ax^2 + bx + c = 0$ can be expressed via Cardano’s formula using radicals of the form: $$x=\sqrt[3]u+\sqrt[3]v+k$$ where $u$ and $v$ depend on the coefficients and the discriminant $\Delta$. ​I am curious about the...

One question from a past qualifying exam is to find the Galois group of $x^6+3$ over the finite field $\mathbb{F}_5$. With some creativity, we can write: $$x^6+3=x^6+8=(x^2)^3+2^3=(x^2+2)(x^4-2x^2+4)=(x^2+2)(x^4+4x^2+4-16x^2)\\ =(x^2+2)((x^2+2)^2-(4x)^2)=(x^2+2)(x^2-2x+2)(x^2+2x+2).$$ Since...

One question in a past qualifying exam asked us to find the Galois group of $x^6+3$ over $\mathbb{F}_5$, $\mathbb{F}_7$, and $\mathbb{Q}$. Each part of this question has been answered individually on this website: over $\mathbb{F}_5$, $\mathbb{F}_7$, and $\mathbb{Q}$. I asked this question again...

Let $E/K$ be a finite field extension, $G=\operatorname{Aut}_K(E)$ be the group of automorphisms fixing elements of $K$, and $E^G$ be the fixed field of $E$ under $G$. We know $E/E^G/K$ and so $$[E:K]=[E:E^G][E^G:K]=|G|\,[E^G:K].$$ Additionally, $[E:K]=[E:K]_s[E:K]_i,$ where $[E:K]_s$ denote the...

A field extension $F/K$ is Galois if $F^{\operatorname{Aut}_K(F)}=K$ in Hungerford's Algebra, and it is Galois if it is normal and separable in Lang's Algebra. For the finite extension, these two definitions are the same. For infinite algebraic extensions or for transcendental extensions, which...

Let $F/K$ be a finite field extension, $G=\text{Aut}_K(F)$ be the group of automorphisms of $F$ that fix elements of $K$, and $F^G$ be the fixed field of $G$. We then have $$[F:F^G]\le |G|.$$ This is proven in Hungerford Chapter V, Lemma 2.9. Hungerford used this lemma to prove "$F^G=K$ iff...

In the proof of Theorem 8.12 in Morandi's Field and Galois Theory, the author writes: Because $KN$ is the composite of a Galois extension of $S$ with a purely inseparable (hence normal) extension, $KN/S$ is normal. Thus, $\sigma_j(K)\subseteq KN$ by Proposition 3.28. I do not understand this...

I was going through the proof of the Abstract Primitive Element Theorem and had minor concerns about how the Galois correspondence is invoked. Abstract Primitive Element Theorem: Let $K$ be an infinite field and let $L/K$ be a finite separable extension. Then there exists $\theta\in\ L$ such...

In chapter 4 of J. P. Serre's "Topics in Galois Theory", he computes the Galois groups of the splitting fields over $\mathbb Q(T)$ of a few polynomials of the form $f(X,T)=f(X)-T$. He does this by calculating their ramification type (i.e. which valuations ramify in this field extension with...

I want to read about inverse Galois Theory with the goal of proving the Hilbert Irreducibility Theorem. I do know the basics of Algebra (Group and Ring Theory, Field and Galois Theory, a bit of Moduls). Is there any good book which is on an undergraduate level about Inverse Galois Theory?

It is a well known fact that the Galois group $G$ of a Galois extension $K\subseteq L$ is a profinite group, as $G$ is equal to the inverse limit of the Galois groups of the finite subextensions of $K\subseteq L$. Therefore, $G$ gets a "natural" topology that turns it into a compact group. Why...