§ Prime numbers as maximal among principal ideals
I learnt of this characterization from benedict gross's lectures, lecture 31.
We usually define a number as prime iff the ideal generated by
, is prime. Formally, for all , if
then or .
This can be thought of as saying that among all principal ideals, the
ideal is maximal: no other principal ideal contains it.
§ Element based proof
- So we are saying that if then either
- Since we can write . Since is prime,and , we have that either .
- Case 1: If then we get . This gives ,or .
- Case 2: Hence, we assume , and .Since , we can write for some .This gives us and . Hence . Thus is a unit, therefore .