F field => F integral domain, so the coefficient of the largest term x^(m + n) will never be zero.
The units are constants.
If is a field then the units in are precisely the units in . True or false?
The answer is not given in the book.
My answer: True.
Proof:
let
where that is degree of and degree of
now, can be written as
.
This gives (doesnt it?)
since is a field one of is 0 contradicting the fact that the degrees of the polynomials were
is this correct?
by the very definition of a field, every non-zero element is a unit, because for any non-zero a, 1/a exists, and we have a(1/a) = 1.
so, yes, the non-zero constant polynomials are the units of F[x], which can be considered "the units of F" through the isomorphism F-->F[x]:
a--->f(x) = a.
(there is a slight distinction between a field element, a, and f(x) = a, which is a "polynomial expression". the constant polynomials of F[x]
"act just like" field elements, and the isomorphism is often understood implicitly).