Abstract Algebra (8): Galois Theory — The Bridge Between Fields and Groups

Wed, 15 Sep 2021 09:00:00 +0000

In 1832, a twenty-year-old mathematician named Evariste Galois, on the eve of a duel he would not survive, wrote down the final versions of his mathematical ideas in a letter to a friend. Those ideas — connecting the symmetries of polynomial roots to the structure of groups — would take more than a decade to be understood and published, but they would reshape algebra forever. Galois theory, as we now call it, establishes a precise dictionary between intermediate fields of a field extension and subgroups of a group of symmetries. It explains, in one elegant framework, why the quadratic formula exists, why there is no analogous formula for degree five, and what “solvability” really means.

Abstract Algebra (7): Field Extensions — Building Bigger Number Systems

Mon, 13 Sep 2021 09:00:00 +0000

Every mathematician, at some point, encounters a polynomial that refuses to be solved within the number system at hand. The ancient Greeks discovered that $\sqrt{2}$ is irrational — that is, $$x^2 - 2$$ has no solution in $\mathbb{Q}$ . The resolution was not to abandon the polynomial, but to enlarge the field. Field extensions formalize this enlargement and give us the structural scaffolding on which Galois theory is built.

I find it useful to think of a field extension as a kind of controlled inflation of a number system. We pump in just enough new elements to solve the equations we care about, and the tower law tells us exactly how much air we used. The resulting picture is much cleaner than I expected when I first met it: every step has a finite degree, the degrees multiply along chains, and the whole thing turns into linear algebra over the base field. This article develops the theory from the ground up: degrees and bases, simple extensions and minimal polynomials, the tower law, splitting fields, and separability. By the end, we will have the full toolkit needed to state and prove the Fundamental Theorem of Galois Theory in the next article.

Field-Theory on Chen Kai Blog

Abstract Algebra (8): Galois Theory — The Bridge Between Fields and Groups

Abstract Algebra (7): Field Extensions — Building Bigger Number Systems