a polynomial of degree over a field has at most roots
Lemma (cf. factor theorem).
Let be a field and let be a non-zero polynomial in of degree . Then has at most roots in (counted with multiplicity).
Suppose that any polynomial in of degree has at most roots and let be a polynomial of degree . If has no roots then the result is trivial, so let us assume that has at least one root . Then, by the lemma above, there exist a polynomial such that:
Hence, is a polynomial of degree . By the induction hypothesis, the polynomial has at most roots. It is clear that any root of is a root of and if is a root of then is also a root of . Thus, has at most roots, which concludes the proof of the theorem. ∎
|Title||a polynomial of degree over a field has at most roots|
|Date of creation||2013-03-22 15:09:01|
|Last modified on||2013-03-22 15:09:01|
|Last modified by||alozano (2414)|