Extensions of what? How are we given the Galois groups? How do we choose to view a given group to construct these extensions?
Let’s say rationals. And some finite group. Start with a basic group like a cyclic group. How do we construct cyclic extensions? How do they look like?
We have to start with some extensions we understand? Take cyclotomic extensions say with pth roots of unity. p-1 is the degree of that extension. If we can view the cyclic group as a quotient of Z/p-1 by a subgroup, then taking the field corresponding to that subgroup we get an extension of Q with the required subgroup. A way to do it is to take a prime p with p\equiv 1 mod n so that we have a subgroup of order p-1/n and the quotient is cyclic of order n.
What about arbitrary abelian groups? By structure theorem, they are products of cyclic groups. By the above, we can create fields whose extensions have Galois groups that correspond to these cyclic groups. But how do we get a field whose Galois is the product? If we can make sure that the fields obtained are essentially disjoint (meaning that the intersections contain only rationals), then the Galois group of the composite of the fields is the product of the cyclic groups. To make sure the fields are disjoint, we use different primes to create them. Say p \equiv 1 mod n and q \equiv 1 mod m are distinct primes. The subgroup with p-1/n elements in Z/p-1 gives a subfield of Q(\zeta_p) and the subgroup with q-1/n elements in Z/q-1 gives a subfield of Q(\zeta_q) – The union of these fields has Galois group $Z/p \times Z/q$
How do I create $S_n$ extensions? What about $A_n$ extensions?Over $\mathbb{Q}$, why should these even exist first?
We need some understanding of Galois groups? What kind of permutations of the roots are allowed? We need transitive subgroups- we should be able to map one root of an irreducible polynomial to another. We can create S_n extensions by making sure that they have some group elements inside the Galois group- like say transposition and n-1 cycle.
How do I compute Galois groups given a polynomial? Look at f(x) mod p and that allows us to find some elements and their cycle structure.
How do we get generic S_n extensions or A_n extensions? Can create extensions with fixed “local properties” – looking at mod p, we have a required Galois group, fixed splitting for some primes or we have fixed number of real/complex roots etc ?
From Galois’s idea of considering permutations of roots that fix all the equations satisfied by the roots to viewing all of it as a group action or as field automorphism- what is the story?
A Neighbourhood Of Infinity 