analytically calculate the inverse of $4^{th}$ order polynomial

Question

There was a quiz test: Question: which is the inverse polynomial of $( X^4 + X + 1)$ ?

a) $(X^3 + X + 1)$

b) $(X^4 + X + 1)$

c) $(X^4 + X^2 + 1)$

d) $(X^4 + X^3 + 1)$

e) the correct answer is missing

and the correct answer was (d).

Could you please show me the steps how to find this inverse function?

question

keys

Answer 1

Notice that

$$X^4\left(\left(\frac1X\right)^4+\left(\frac1X\right)+1\right)=X^4+X^3+1$$

and that in GF(2), the additive inverse of

$$X^4+X+1$$ is itself.

Answer 2

Perhaps there is some mis-understanding. The inverse of the function $y=x^4+x+1$ is $x$ as a function of $y$ and that is a complicated multi-valued $4$-cycled algebraic function in terms of $y^\frac{1}{4}$. You can get some idea of it's complexity by simply solving for the inverse in Mathematica via the command:

Solve[y==x^4+x+1,x]

Also, we can numerically invert it by (numerically) solving the associated monodromy differential equation. To get some idea of what that involves see, Algebraic Functions