Rings, ideals, fields and polynomials

By the end you’ll be able to

• Identify rings, subrings, ideals, integral domains and fields, including $\mathbb{Z}_n$, polynomial rings, Gaussian integers and matrix rings
• Find units, zero divisors and the characteristic of a ring, and decide when $\mathbb{Z}_n$ is a field
• Test polynomials for irreducibility over $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{Z}_p$ (roots for degree 2 and 3, Eisenstein’s criterion, reduction mod $p$) and describe quotient rings $F[x]/(p(x))$