Answer to Question #284663 in Abstract Algebra for Bret

Question

Show that the polynomial 2x + 1 in Z4[x] has a multiplicative inverse in Z4[x].

Answer 1

Solution:

Assume $(1+2x).(1+2x)$

$(1 + 2x)^2 = 1 + 4x + 4x^2 = 1.$

The inverse of $2x + 1$ in $Z_4[x]$ is $2x + 1$.

Thus, the polynomial $2x + 1$ in $Z_4[x]$ has a multiplicative inverse in $Z_4[x]$.