Course notes (chap. 1 Number Theory, chap. 2 ... - McGill University

2.2 Polynomials over a field
ApolynomialoverF p is specified by a finite sequence (a n ,a n−1 ,...,a 1 ,a 0 )of
elements from F p ,witha n ≠0.Thenumbern is the degree of the polynomial.
We have operations +, −, × on polynomials analogous to the similar integer
operations. Addition and subtraction are performed componentwise using
the addition + and subtraction − of the field F p .
Products are computed by adding all the products of coefficients associated
to pairs of exponents adding to a specific exponent.
Example:
(x 4 + x +1)× (x 3 + x 2 + x)
= x 4 × (x 3 + x 2 + x)+x × (x 3 + x 2 + x)+1× (x 3 + x 2 + x)
= (x 7 + x 6 + x 5 )+(x 4 + x 3 + x 2 )+(x 3 + x 2 + x)
= x 7 + x 6 + x 5 + x 4 +(1+1)x 3 +(1+1)x 2 + x
= x 7 + x 6 + x 5 + x 4 + x