# Group of Units

• Jan 19th 2011, 07:14 PM
mulaosmanovicben
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?
• Jan 19th 2011, 07:57 PM
roninpro
Everything looks good except (c). You are missing a lot of units. For example, what about $2$? It has an inverse: $1/2$.
• Jan 19th 2011, 08:10 PM
mulaosmanovicben
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?
• Jan 19th 2011, 09:26 PM
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.
• Jan 20th 2011, 06:02 AM
topspin1617
Quote:

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).