Concept

Galois group

The group of symmetries of the roots of a polynomial that preserve every relation among them with rational coefficients. Its subgroups match the fields between the rationals and the roots, which is how it decides what can be solved by radicals.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The subgroups of a polynomial's symmetries, against the fields they name. Two lattices side by side, one of the subgroups of the symmetry group of the cube roots of two and the other of the fields between the rationals and the splitting field, drawn so that one is the other turned upside down.

The lattice that runs the other way

The symmetries of a polynomial's roots form a group, and the fields between the bottom and the top form a lattice. The two are the same picture, one of them turned over — a bigger group of symmetries fixes less, so it names a smaller field.

algebra · Galois correspondence

The three pairings of the roots of x⁴ + x + 1. Three panels each showing the same four roots of a quartic in the complex plane, joined in a different way into two pairs, with the value of the sum of the pair products beneath.

Three ways to pair four roots

Four roots can be split into two pairs in exactly three ways, and the three numbers r·r′ + r″·r‴ those pairings give are the roots of a cubic whose coefficients can be read straight off the quartic. That cubic is where Ferrari's formula gets its cube roots, and it is also a verdict: whether its roots are rational decides which of the twenty-four symmetries the quartic's roots actually have.

algebra · Galois correspondence

Named alongside it

The objects these essays reach for when they reach for this one.

Automorphism Degree Discriminant Field extension Fixed field Index Lattice Normal subgroup Permutation Polynomial Quotient Roots of unity

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