Thread: Group of Units

I am not sure exactly what this question is asking:

Determine the following groups of units:

a) U(Z[x]) where Z[x] is all polynomials with integer coefficients
b) U(Z[i]) where Z[i] is the gaussian integers (a+bi)
c) U(R[x]) where R[x] is all polynomials with real coefficients

I think this means find which elements in these groups have inverses
I am not quite sure but I have:

a) 1,-1
b) 1,-1,i,-i
c) 1,-1

Thoughts?

Everything looks good except (c). You are missing a lot of units. For example, what about ? It has an inverse: .

Okay, so for


C) I should also include all of R (real numbers) as a units (for ever a in R 1/a is its inverse) is that all? Is it possible to determine the inverse of a polynomial?

That's all of them. If you think about it, there is no way to invert a polynomial. No matter what you multiply it by, the degree will either stay the same or get larger; you will never be able to make it 1.

Originally Posted by roninpro

That's all of them. If you think about it, there is no way to invert a polynomial. No matter what you multiply it by, the degree will either stay the same or get larger; you will never be able to make it 1.

Just to clarify for anyone coming across this (even though it doesn't really matter in this PARTICULAR problem), this is only true when the coefficient ring is an integral domain (which it is in both polynomial rings in this problem).