multiplicative inverse from irreducible polynomial

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If you ask about computing multiplicative inverse modulo an irreducible polynomial, then this can be done by defining a corresponding quotient ring. For example, the following code inverts $x+1$ modulo $x^4+x+1$ over the field $GF(2)$:

R.<x> = PolynomialRing(GF(2))
K.<a> = R.quotient_ring(x^4+x+1)
print( (a+1)^(-1) )

Here a stands for a zero of $x^4+x+1$, which also may be thought of as the image of $x$ in the quotient ring.