Polynomial ring

A polynomial is a formal expression: two polynomials are equal exactly when all their coefficients agree. This sets it apart from the function it defines. In \(\mathbb{Z}_2[x]\), the polynomials \(x^2\) and \(x\) are different, even though as functions they agree at \(0\) and \(1\). That is why we write \(R[x]\) and speak of the indeterminate \(x\), not a variable.

The degree of a nonzero polynomial is the largest exponent with nonzero coefficient. Always \(\deg(f+g)\le\max(\deg f,\deg g)\), and if the leading coefficient of \(f\) times that of \(g\) is nonzero, then \(\deg(fg)=\deg f+\deg g\). This fails when there are zero divisors: in \(\mathbb{Z}_6[x]\), \((2x+1)(3x+1)=5x+1\) has degree 1, not 2.

Properties of \(R\) pass to \(R[x]\): if \(R\) is commutative, so is \(R[x]\); if \(R\) is an integral domain, so is \(R[x]\), and its units are those of \(R\). When \(R\) is not a domain there can be unexpected units: in \(\mathbb{Z}_4[x]\), \(1+2x\) is its own inverse. The ring \(R\) is identified with the constant polynomials, and \(R[x,y]=R[x][y]\) builds several indeterminates.

A polynomial ring is never a field even if \(R\) is one: \(x\) has no inverse, by the degree argument. What does hold, when \(R=F\) is a field, is a division algorithm, the topic of the next concept, and with it polynomials inherit an arithmetic similar to that of \(\mathbb{Z}\). Also, evaluation \(f\mapsto f(a)\) is a ring homomorphism, to which the first isomorphism theorem applies.