Roots and Irreducible Polynomials

This handout, which accompanies the course on analogies between Z and F [T ], discusses some properties of polynomials in F [T ]. The results in Sections 1 and 2 work with any F , but the results in Section 3 and 4 are (somewhat) special to the field F = Fp. The main result in these notes in Theorem 3.7. The notation F [T ]h(T ), for the ring of polynomials in F [T ] considered modulo h(T ), is used on the first-year number theory sets. We will instead use a more common notation from abstract algebra, and write F [T ]/h(T ) for F [T ]h(T ). As a matter of terminology, when f(T ) ∈ F [T ], we will say f(T ) is a polynomial “over” F . For example, if we are thinking about T 3 +2T +1 as a polynomial in F5[T ] (as opposed to F2[T ], R[T ], and so on), we might say “consider T 3 + 2T + 1 over F5.”