Factor Rings of Polynomials Over a Field
On page 222 of Nicholson: Introduction to Abstract Algebra in his section of Factor Rings of Polynomials Over a Field we find Theorem 1 stated as follows: (see attached)
Theorem 1. Let F be a field and let be an ideal of F[x]. Then a uniquely determined monic polynomial h exists exists in F[x] such that A = (h).
The beginning of the proof reads as follows:
Proof: Because , it contains non-zero polynomials and hence contains monic polynomials (being an ideal) ... ... etc. etc.
BUT! why must A contain monic polynomials??
Help with this matter would be appreciated!
Re: Factor Rings of Polynomials Over a Field
Since A is an ideal, is monic and is in A.