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?




closed as unclear what you're asking by Yves Daoust, Dietrich Burde, Lord Shark the Unknown, Shailesh, Martin Argerami May 8 at 3:38

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    $\begingroup$ Have you tried multiplying the given answer with your polynomial?^^ $\endgroup$ – Riquelme May 7 at 11:31
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    $\begingroup$ @nauok . Would you mind to write clearly (in your question, not in comments) your definition of the "inverse of a polynomial". If not I am afraid that the downvotes will pour down. $\endgroup$ – JJacquelin May 7 at 11:51
  • $\begingroup$ @JJacquelin I edited it. is there a way you can suggest? $\endgroup$ – nauok May 7 at 11:59
  • $\begingroup$ You must have misunderstood (or misreproduced) the homework. The functional inverse of the given polynomial is not a polynomial (in fact, it is not invertible). $\endgroup$ – Yves Daoust May 7 at 12:02
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    $\begingroup$ What is the exact wording of the quiz? Please note that there is something called reciprocal polynomial (or reflected polynomial) which have nothing to do with inverse of a polynomial... $\endgroup$ – achille hui May 7 at 12:16

Notice that


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

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

  • $\begingroup$ Thank you. so all i had to do was to factor x^4 out? $\endgroup$ – nauok May 7 at 12:40
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    $\begingroup$ I don't think you understood my answer. It shows that with crazy interpretations, b) or d) hold. $\endgroup$ – Yves Daoust May 7 at 12:52
  • $\begingroup$ AH. OK. i give up. $\endgroup$ – nauok May 7 at 13:00

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:


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


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