Projective Geometry I - Lecture Notes

7.1 Division Rings

Def: Ring, Division Ring, Field

Division Rings are also called skewfields.

Examples: Z is a commutative ring.

Mn(any ring), non-commutative and not every element has an inverse.

Q, R ,C are fields.

Def: Subfield

Example: Zn is a commutative ring. Zp is a field.

Def: Characteristic of a division ring.

Proposition 7.7: The characteristic of a division ring is either 0 or a prime.

Ex: Q,R,C have characteristic 0. Zp has characteristic p.

7.2 The Quaternions H

Def: H, the quaternions, give an example of an infinite skewfield of characteristic 0.

Def: The center of a division ring.

Proposition 7.11: The center of a division ring is a field.

Def: Automorphism of a division ring.

Ex: inner automorphism. Every automorphism of H is inner.

6.3 A noncommutative Division Ring with Characteristic p

Proposition 7.14: There exists a noncommutative division ring of arbitrary characteristic p.

The skew Laurent series ring in one indeterminate.