I get it - I think. Nearly finished Stewrart's book - not properly worked through some of the later chapters but have read them fairly carefully, and followed (most of) the proofs.
The gist of it is a correspondence between a group (the Galois group formed from automorphisms of a field) and extensions of the field. If the field of rational numbers (Q) is extended by adjoining the element i, the automorphisms of Q are the identity and complex conjugation (a+bi maps to a-bi). These form a group isomorphic to Z2. With more complex field extensions you get more complex groups. The correspondence is one-one if the field extension is both normal and separable.
The roots of a polynomial with coefficients in Q probably won't be in Q because they will involve nth roots. Probably the most significant result from Galois theory is that a polynomial is solvable by radicals if and only if the Galois group of the field extension that includes the roots is soluble. For a group to be soluble if has to have a chain of normal subgroups such that adjacent quotient groups are Abelian. The Galois group of the general polynomial of degree n is the permutation group Sn. Quadratics, cubics and quartics are soluble because their Galois groups are soluble. The general quintic equation is not solvable as S5 is not soluble. It has one normal subgroup, the alternating group A5, which itself is simple (has no normal non-trivial proper subgroups). So that's why you can't solve a general quintic.
I've still got a bit of work to do yet with a few exercises, and a bit more study of the more abstract Galois theory in finite fields, but overall, I think I get it.
Galois theory really is a great piece of classical mathematics, giving the answers to the celebrated problems of antiquity (squaring the circle - not possible because pi is transcendental and you can only construct fields extensions of order 2n using ruler and compasses), polynomial solutions and the fundamental theorem of algebra. It's also a great way of consolidating group theory knowledge and learning about other algebraic structures - rings, fields, integral domains and vector spaces are all there. I highly recommend anyone who knows maths to read up on it.
Galois Theory
I get it - I think. Nearly finished Stewrart's book - not properly worked through some of the later chapters but have read them fairly carefully, and followed (most of) the proofs.
The gist of it is a correspondence between a group (the Galois group formed from automorphisms of a field) and extensions of the field. If the field of rational numbers (Q) is extended by adjoining the element i, the automorphisms of Q are the identity and complex conjugation (a+bi maps to a-bi). These form a group isomorphic to Z2. With more complex field extensions you get more complex groups. The correspondence is one-one if the field extension is both normal and separable.
The roots of a polynomial with coefficients in Q probably won't be in Q because they will involve nth roots. Probably the most significant result from Galois theory is that a polynomial is solvable by radicals if and only if the Galois group of the field extension that includes the roots is soluble. For a group to be soluble if has to have a chain of normal subgroups such that adjacent quotient groups are Abelian. The Galois group of the general polynomial of degree n is the permutation group Sn. Quadratics, cubics and quartics are soluble because their Galois groups are soluble. The general quintic equation is not solvable as S5 is not soluble. It has one normal subgroup, the alternating group A5, which itself is simple (has no normal non-trivial proper subgroups). So that's why you can't solve a general quintic.
I've still got a bit of work to do yet with a few exercises, and a bit more study of the more abstract Galois theory in finite fields, but overall, I think I get it.
Galois theory really is a great piece of classical mathematics, giving the answers to the celebrated problems of antiquity (squaring the circle - not possible because pi is transcendental and you can only construct fields extensions of order 2n using ruler and compasses), polynomial solutions and the fundamental theorem of algebra. It's also a great way of consolidating group theory knowledge and learning about other algebraic structures - rings, fields, integral domains and vector spaces are all there. I highly recommend anyone who knows maths to read up on it.