Let K be a subfield of C and f,g $\in$ K[x]. Let z be a complex root of f and assume f is irreducible in K[x] and g(x)-z is irreducible in K[z][x]. Then f(g(x)) is irreducible in K[x]
we want additive identity and multiplicative identity to exist, i.e. a number we call 0 such that x+0=x for all x, and a number we call 1 such that x*1=x for all x
formal (adj) of or concerned with outward form or appearance as distinct from content. "I don't know enough about art to appreciate the purely formal qualities"
@AlexKChen but here we are not considering them as functions
if we treat them as functions, we start running into problems like "why is 0 and x^2+x different polynomials over Z/2Z if they agree at every point in the domain?"
because a function is uniquely determined by its domain, codomain, and the image of each point in the domain
A nontrivial example is the following, having defined K[X] you can define K[X,Y] as (K[X])[Y], that is polynomials in Y with coefficents in K[X] (and inductively you can define the ring of polynomials in any number of finite variables)
for the purposes now, a homomorphism is any function that preserves those things, a monomorphism is an injective homomorphism, an epimorphism is a surjective homomorphism, and an isomorphism is a bijective homomorphism
