U(Z[x]) ?

1800bigk

Hi, I have to find the units of Z[x]. I am little unclear and my book does not go into detail. Does U(Z[x]) mean I need to find every polynomial with integer cooefficients that has a multiplicative inverse? or do I have to find the multiplicative identity? I was thinking about both cases and the number of polynomials with a multiplicative inverse is pretty limited, isn't it? f(x)=1 or f(x)=-1. As for inverses of polynomials, there would be none because if you multiply a polynomial with x by another polynomial with x then the powers of x get bigger.

Hurkyl

It means to find all elements of Z[x] with a multiplicative inverse. (That's what it means to be a unit)

That sounds plausible... can you work it into a rigorous proof?

1800bigk

am I right about f(x)=1 and f(x)= -1 being the only polynomial in U(Z[x]) or is there more? would contradiction be the best way, how would I start it?

tia

Hurkyl

I generally hate answering this question. Learning how and when to be confident in your own work is important! But yes, you are correct.

There are lots of ways. I would suggest starting with a literal translation of what you said:

If p(x) and q(x) are nonconstant, then p(x)*q(x) is nonconstant.

(Actually, you made a stronger statement, but I don't want to spoil figuring out how to translate that!)

And see if you can prove this statement and relate it to inverses.

1800bigk

ok thanks!