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Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where $k$ is divisible by two, such as say $k=8$, or $k=12$.

If so, can they be constructed easily? I'd be grateful if you could provide a few examples.

clan
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    These units are represented by the polynomials in $\mathbf{F}_2[x]$ of degree less than $k$ which are coprime to $x^k+1$. You can certainly compute these for any particular $k$, as long as $k$ is small enough that you'll have time to write down the answer. You might get more detailed answers if you posted this to math.stackexchange. – Michael Zieve Dec 06 '13 at 14:54
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    In particular, for $k=8$, $x^8+1=(x+1)^8$, and the units are the polynomials with an odd number of nonzero terms. – Derek Holt Dec 06 '13 at 14:56
  • Thanks for the comments. I've just posted the question to math. Obviously if the polynomial is not co-prime to $x^k+1$ it's a zero divisor. Am I guaranteed that otherwise it's a unit? I guess $k=12$ is the harder case, since it's not a power of $2$. – clan Dec 06 '13 at 15:09

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