This is an introductory lecture, giving an informal overview of Galois theory. We discuss some historical examples of problems that it was used to solve, such as the Abel-Ruffini theorem that degree 5 polynomials cannot in general be solved by radicals, and Wiles's proof of Fermat's last theorem. The classic book "Galois theory" by E. Artin has been reprinted by Dover and is strongly recommended.
Playlist - Galois theory (Borcherds)
We review some basic results about field extensions and algebraic numbers.
We define the degree of a field extension and show that a number is algebraic over a field if and only if it is contained in a finite extension. We use this to show that the sum and product of algebraic numbers is algebraic, and that a root of a polynomial with algebraic coefficients is algebraic.
We define the splitting field of a polynomial p over a field K (a field that is generated by roots of p and such that p splits into linear factors). We give a few examples, and show that it exists and is unique up to isomorphism.
We define the algebraic closure of a field as a sort of splitting field of all polynomials, and check that it is algebraically closed.
We then give a topological proof that the field ℂ of complex numbers is algebraically closed, and mention the field of formal Puiseaux series as another example of an algebraically closed field.
Finally we discuss the uniqueness of algebraic closures of fields, in particular the fact that although any two algebraic closures are isomorphic, the isomorphism is not unique. We mention the analogy with the problem of defining the fundamental group of a topological space.
We use the theory of splitting fields to classify finite fields: there is one of each prime power order (up to isomorphism).
We give a few examples of small order, and point out that there seems to be no good choice for a standard finite field of given order: this depends on the choice of an irreducible polynomial.
Finally we show how to count the number of irreducible polynomials of given degree in a finite field (using the field of order 64 as an example).
We define normal extensions of fields by three equivalent conditions, and give some examples of normal and non-normal extensions. In particular we show that a normal extension of a normal extension need not be normal.
We define separable algebraic extensions, and give some examples of separable and non-separable extensions. At the end we briefly discuss purely inseparable extensions.
We define Galois extensions in five different ways, and show that four of these conditions are equivalent. (The 5th equivalence will be proved in a later lecture.) We use this to show that any finite group is the Galois group of some finite extension.
We give several examples of Galois extensions, and work out the correspondence between subfields and subgroups explicitly.
We prove the main theorem of Galois theory, given a bijection between subgroups of a Galois group and subextensions of a Galois extension. We illustrate it with the example of the splitting field of 4th roots of 2.
As an application of Galois theory, we prove Gauss's theorem that it is possible to construct a regular heptadecagon with ruler and compass.
We use Galois theory to give a (mostly) algebraic proof that the complex numbers form an algebraically closed field.
We show that any finite separable extension of fields has a primitive element (or generator) and given n example of a finite non-separable extension with no primitive elements.
We discuss Abel's theorem, that says a general quintic equation cannot be solved by radicals. We do this by showing that if a polynomial can be solved by radicals over a field of characteristic 0 then its roots lie in a solvable Galois extension. We give some examples of degree 5 polynoimals whose roots do not generate a solvable extension.
We describe Galois extensions with cyclic Galois group of order n in the case when the base field contains all nth roots of unity and has characteristic not dividing n. We show that all such extensions are radical. As an example we express cos 2π/7 explicitly using cube roots. Finally we mention the Kummer pairing.
As an application of Galois theory, we prove Gauss's theorem that it is possible to construct a regular heptadecagon with ruler and compass.
We show how to use Galois theory to solve cubic and quartic polynomials by radicals.
We show that the Frobenius automorphism of a finite field an sometimes be lifted to characteristic 0. As an example we use the Frobenius automorphisms of ℚ[i] to prove that -1 is a square mod an odd prime p if and only if p is 1 mod 4.
We introduce cyclotomic polynomials, and use Frobenius automorphisms to show they are irreducible. We give two applications of them: we prove a special case of Dirichlet's theorem on primes in arithmetic progressions, and we show that any finite abelian group is the Galois group of a finite extension of the rational numbers ℚ.
We prove Wedderburn's theorem that all finite division algebras are fields. The proof uses cyclotomic polynomials.
We define the norm and trace of a finite extension of fields. We give some examples of calculating the image of the norm map, and show how to use the norm and trace to find rings of algebraic integers.
We define the discriminant of a finite field extension, and show that it is essentially the same as the discriminant of a minimal polynomial of a generator. We then give some applications to algebraic number fields.
We discuss two forms of Hilbert's theorem 90: the original version for cyclic extensions, and Noether's more general version for arbitrary finite Galois extensions. The proofs use a lemma of Artin about the linear independence of group characters.
We show how to extend Galois theory to infinite Galois extensions. The main difference is that the Galois group has a topology, and intermediate field extensions now correspond to closed subgroups of the Galois group. We give some examples, such as the absolute Galois group of a finite field, and the Galois group of the cylotomic extension of the rationals.
We also show that the Gaussian integers cannot be extended to a Galois extension with Galois group ℤ/4ℤ, which put some restrictions on the absolute Galois group of the rationals.
We describe transcendental extension of fields and transcendence bases. As applications we classify algebraically closed fields and show how to define the dimension of an algebraic variety.
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