Finally, I feel I have a little grip on Galois Theory (GT) that I thought I should document my understanding at a very broad level. I think central to the concept of gaining a high-level intuition on GT are two essential ideas - (a) the notion of a Galois group and (b) the Galois correspondence. In this blog, we will see what is a Galois group.
In the following let us assume that we know the concepts of fields, extension and base fields, polynomials, and symmetric group Sn on n symbols, which are essentially permutations.
(a) The first key idea is about the concept of an automorphism. What is an automorphism in the context of GT?
An automorphism φ: E→E on the extension field E that "fixes" the base field Q is a map that preserves the field structure, but accomplishes more than that because of the "fixing" property. For our purposes, given a polynomial that has its roots in E, it is essentially a permutation of the roots of the polynomial. To see why this true,
Let P(x) = ∑aixi be a polynomial with r as a root. Then,
0 = φ(0)
= φ(∑airi)
= ∑aiφ(r)i because φ(ai) = ai by fixing property and φ is an homomorphism.
Therefore, φ(r) is also a root and the mapping r → φ(r) is essentially a permutation. Thus, the set of automorphisms will be a set of permutations that will form a group under composition. This group is usually referred to as the Galois group denoted by Gal[E/Q], and for all practical purposes, we can think of Gal[E/Q] as a subgroup of the symmetric group of permutations Sn.
In the next blog, I will illustrate Galois correspondence, the other name for The Fundamental Theorem of Galois Theory.
(a) The first key idea is about the concept of an automorphism. What is an automorphism in the context of GT?
An automorphism φ: E→E on the extension field E that "fixes" the base field Q is a map that preserves the field structure, but accomplishes more than that because of the "fixing" property. For our purposes, given a polynomial that has its roots in E, it is essentially a permutation of the roots of the polynomial. To see why this true,
Let P(x) = ∑aixi be a polynomial with r as a root. Then,
0 = φ(0)
= φ(∑airi)
= ∑aiφ(r)i because φ(ai) = ai by fixing property and φ is an homomorphism.
Therefore, φ(r) is also a root and the mapping r → φ(r) is essentially a permutation. Thus, the set of automorphisms will be a set of permutations that will form a group under composition. This group is usually referred to as the Galois group denoted by Gal[E/Q], and for all practical purposes, we can think of Gal[E/Q] as a subgroup of the symmetric group of permutations Sn.
In the next blog, I will illustrate Galois correspondence, the other name for The Fundamental Theorem of Galois Theory.
