We prove that Z is universal in the category of rings with identity. Using that, we define the characteristic of a field, and prove that it is well-defined. We then discuss ring ideals a bit more before finishing by showing that all fields with exactly p elements (where p is prime) are isomorphic.
We introduce localization of a ring and the ring of fractions of a ring. We then prove that the ring of polynomials over a field is a Euclidean Domain, and then finish by proving Gauss’ Lemma that a polynomial is irreducible over a ring exactly when it is irreducible over the ring’s field of fractions.
We’ve already introduced some of the topics of Group Theory. In this entry we’ll talk about Lagrange’s Theorem, various morphisms, and try to tie it back to our ongoing discussion of permutations. We’ll also give some examples of groups…
