Definition (ring extension):
Whenever is a ring and is a subring of , we say that is a ring extension of and write .
Note that if is a ring extension, then is a ring extension; indeed, the set is the set of all polynomials with coefficients in , the set is the set of all polynomials with coefficients in , and is a subring of .
Proposition (existence of splitting ring):
Let be a ring, and let be a polynomial over . Then there exists a ring extension such that in , decomposes into linear factors, that is,
- for certain .
Proof: We prove the theorem by induction on the degree of . Suppose first that can be decomposed into two polynomials