quick question on rings of polynomials.

If http://latex.codecogs.com/png.latex?F is a field then the units in http://latex.codecogs.com/png.latex?F[x] are precisely the units in http://latex.codecogs.com/png.latex?F. True or false?

The answer is not given in the book.

My answer: True.

Proof:

let http://latex.codecogs.com/png.latex?f(x),g(x) \in F[x]

where http://latex.codecogs.com/png.latex?...+\ldots+b_mx^m that is degree of http://latex.codecogs.com/png.latex?f \, is \, n>0 and degree of http://latex.codecogs.com/png.latex?g \, is \, m>0

now, http://latex.codecogs.com/png.latex?f(x)g(x) can be written as

http://latex.codecogs.com/png.latex?...+a_nb_mx^{m+n}.

http://latex.codecogs.com/png.latex?..._nb_mx^{n+m}=1

This gives http://latex.codecogs.com/png.latex?a_nb_m=0 (doesnt it?)

since http://latex.codecogs.com/png.latex?F is a field one of http://latex.codecogs.com/png.latex?a_n\, or\, b_m is 0 contradicting the fact that the degrees of the polynomials were http://latex.codecogs.com/png.latex?n \,and\, m

is this correct?