# What techniques can be used to show that a multiplicative inverse doesn't exist? [duplicate]

Example: To show that the polynomials of a finite field is itself not a field, I need to show that a multiplicative inverse does not exist.

What are the general techniques that can be used to show that some ring does not have a multiplicative inverse?

I'm thinking the best way to go about it would be to assume that a multiplicative inverse exists by the definition, and then look for a contradiction.

What types of contradictions could help show that a multiplicative inverse doesn't exist?

Or are there any other ideas?

• In the case of the ring of polynomials in $n$ variables, over any ring, you can determine quite easily whether an element is invertible or not (look at exercise 1.2 and 1.3 on Atiyah-Macdonald) Feb 24 at 11:04
• See the dupe for the case of polynomial rings (the question is far too broad for general rings). Feb 24 at 11:35

Let's consider $$\mathbb{Z}_2[X]$$ and let $$P(X)=X$$. If this polynomial has an inverse $$g(X)=\sum_{i=0}^{\infty}a_iX^i$$ in $$\mathbb{Z}_2[X]$$, then $$P(X)G(X)=1=X(\sum_{i=0}^{\infty}a_iX^i)=\sum_{i=0}^{\infty}a_iX^{i+1}.$$
However, it is impossible that $$\sum_{i=0}^{\infty}a_iX^{i+1}=1$$ as this doesn't even contain a constant term.
There is a very general criterium that you can deduce here. Try the same with $$P(X)=1+X$$, is that one invertible?