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...