MATH 3231 Class Notes 03-14-2025

notecommutative-algebrafield-theory

Defined Frobenius Morphism

For commutative ring $R$, $\mathrm{char}R=p\ge 2$, $p$ prime. There exists a morphism of rings $F$ (the Frobenius morphism) given by,

$$F:R\to R\text{ such that }F(r)=r^p.$$

If $R$ is a domain, then $F$ is injective.

Binomial Formula

For any commutative ring, we have the binomial formula.

Defined Formal Derivative

For a commutative ring $R$, we have $P(X)=a_n{X}^n+\dots$

Theorem

For $p\ge 2$ prime, $e\ge 1$, $q=p^e$. There exists a field with $q$ elements.