Show that $f(x)=2x+1$ has a multiplicative inverse in $\mathbb{Z}_4[x]$, the integers mod 4.
I don't really know where to start (besides dividing $1$ by $f(x)$). I thought having multiplicative inverses was one of the requirements for something to be a field.
Also, if the inverse is just $\frac{1}{2x+1}$, where do I go from there? How do I show that the inverse is in $\mathbb{Z_4}[x]$.
Best Answer
$(2x+1)(-2x+1)=1-4x^2=1$ as polynomials in $\mathbb{Z}_4[x]$. You can check Atiyah-MacDonald (chap.1- ex. 2.i) for a characterizazion of invertible elements in the ring $A[x]$ where $A$ is a generic commutative ring.