Franklin's Notes


Types of primes

In a ring $R$, a prime $p\in R$ is an element such that when $p|ab$, we have that either $p|a$ or $p|b$, for any $a,b\in R$. Alternatively, the principal ideal generated by $p$ is a prime ideal . If $S\supset R$ is a super-ring of $R$ and is also a UFD , then the primes $p\in R$ can be classified as follows:

1. If $p$ is also prime in $S$, we say that it is inert or remains prime.
2. If $p$ is composite in $S$ and some prime factor of its factorization in $S$ has multiplicity $>1$, then we say that $p$ ramifies.
3. If $p$ is composite in $S$ and no prime factor of its factorization in $S$ has multiplicity $>1$, then we say that $p$ splits or is split.

For example, in $\mathbb Z[i]\supset\mathbb Z$, we have that $2=(1+i)(1-i)=i(1-i)^2$ ramifies, and $3$ is inert, and $5=(2-i)(2+i)$ splits.

We can often determine using geometric intuition whether certain primes in quotients of polynomial rings will be inert, ramified, or split. I'll add more on this later, and hopefully a pretty picture.

abstract-algebra

number-theory

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