Fundamental Theorem of Galois Theory Explained

Fundamental Theorem of Galois Theory Explained

Evariste Galois was born in 1811 and was a brilliant mathematician. At the age of 10, he was offered a place at the College of Reims, but his mother preferred to homeschool him. He initially studied Latin when he was finally allowed to go to school, but became bored with it and focused his attention on mathematics.

Galois’ work involving the necessary and sufficient condition for a polynomial to be solved by radicals solved a 350-year-old problem that had been holding the field of mathematics back. This work helped to lay down the foundation of group theory, abstract algebra, and Galois Theory.

At the age of 20, he suffered wounds during a duel which would take his life, yet his contribution to mathematics continues to live on today.

The fundamental theorem of Galois theory comes from mathematics and is a result which describes the structure of certain field extensions. The most basic format of this theorem provides and assertion that if a field extension is finite and Galois, the intermediate fields and the subgroups of the Galois group will have a one-to-one correspondence.

Galois theory maintains that if E is a given field and G is a finite group of automorphisms of E and they are with a fixed field (F), then E/F becomes a Galois extension.

Description of the Correspondence

When dealing with finite extensions, the fundamental theorem of Galois theory is described like this.

1. If a subgroup (H) of Galois, which is E/F, the corresponding fixed field will be denoted as H over E and will be the set of the elements where E are fixed by each automorphism that is in H.

2. For an intermediate field (K) of Galois, still E/F, the corresponding subgroup becomes Aut(E/K), which means the set of autmorphisms in Galois (designated Gal(E/F)) will fix every element of K.

Within the fundamental theorem, correspondence can only be one-to-one communication only if E/F is a Galois extension. If it is not a Galois extension, then the correspondence can only provide an injective map from the subgroups and subfields. It will also provide only a surjective map in the opposite direction.

Properties of the Correspondence

The fundamental theorem of Galois theory provides three specific useful properties.

  • It is inclusion reversing. For example: if the inclusion of the subgroups H1 ⊆ H2 is able to hold, it is because the inclusion of the fields E1 ⊆ E2 is able to hold.
  • The degrees of extensions are directly related to the orders of the groups. It must be in a manner that is consistent with the inclusion reversing properties. For example: if H is a subgroup of E/F as a Galois extension, then |H| = [E:EH] and |Gal(E/F)/H| = [EH:F] must hold true.
  • EH is a field which is a normal extension of F or a Galois extension by definition, but only if H is a normal subgroup of E/F as a Galois extension. That means EH induces an isomorphism due to the restriction of elements within the Gal(E/F) between it and the quotient group.

What is a normal extension?
When operating in abstract algebra, a field extension is defined as being normal if every irreducible polynominal either has no root or splits into linear factors when dealing with L. It is an extension that is very similar to a Galois extension, with its own examples and counterexamples that contribute to the fundamental theorem described here.

What is a normal subgroup?
A normal subgroup is invariant under conjugation by the members of the group to which it belongs. The left and right cosets should coincide. Only normal subgroups can be used to construction a quotient group from any given group.

Applications of the Fundamental Theorem of Galois Theory

The fundamental theorem classifieds the intermediate fields (E/F) with regards to group theory. The translation between the subgroups and the intermediate fields shows that a general quantic equation cannot be solved by radicals. One must first be able to determine the Galois groups of radical extensions and then use the fundamental theory to show that the solvable extensions are able to correspond to the solvable groups.

The fundamental theorem can also be applied to infinite extensions which are normal and separable, if specific topological structures (Krull topology) are defined on the Galois group and only subgroups that are also closed sets are relevant with the correspondence.

Galois theory helped to establish a new era in mathematics and allows for an accurate approach to abstract algebra.