Chapter 10

Galois Theory

In most sciences, one generation tears down what another has built and what one has established another undoes. In mathematics alone, each generation adds a new storey to the old structure.

—Hermann Hankel

The moving power of mathematical invention is not reasoning but imagination.

—Augustus de Morgan

If E ⊇ F is an extension of fields, Galois theory studies the set of automorphisms σ : E → E that fix F in the sense that σ(a) = a for all a F . The set G of all such automorphisms is a group called the Galois group of E over F . With appropriate restrictions on the extension E ⊇ F, we can establish a bijection (called the Galois correspondence) between the subgroups of G and the subfields of E that contain F . This correspondence is very useful in deducing properties of the subfields from properties of the corresponding subgroups and conversely.^{110}

The origins of Galois theory lie in the theory of equations. Methods implying the quadratic formula for solving x^{2} + bx + c = 0 were known to the Babylonians in 1600 BC, but an algebraic formulation did not appear until the second century AD. As to cubics, nothing appears to have been done until the fifteenth century when Scipione del Ferro, and later Niccolò Tartaglia, found what is now called the cubic formula. This result, together with Lodovico Ferrari's formula for solving quartics, was published in 1545 in ...