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If I need to find the multiplicative inverse of an element in some $T[x]/(m)$ factor ring, do I need to solve a diophantine equation to get the solution?

Let the element be $f$. Then $fu \equiv 1$ (mod $m$) so $fu-mv=1$. Is this correct? $u$ is the inverse

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  • It basically amounts to the working through the reverse Euclidean algorithm for polynomials (pretty much the same process one would find the multiplicative inverse of an integer $m \in \mathbb Z/n\mathbb Z$. (What you've said so far is correct). – ah11950 Apr 12 '14 at 11:10
  • @user128576 what you wrote is correct, but knowing something about T might provide a shortcut. Anything special about T? – rschwieb Apr 12 '14 at 11:13
  • it might be C or some Zp – user128576 Apr 12 '14 at 11:19
  • As you've noted, computing an inverse in $\,T[x]/m\,$ is equivalent to solving said diophantine equation in $\,T[x].\,$ If $\,T$ is field then $\,T[x]\,$ has a division algorithm so Euclidean algorithm for the gcd, so the extended Euclidean Division Algorithm may be used to compute the inverse ([see here](http://math.stackexchange.com/a/85841/242) for a convenient method to do so). – Bill Dubuque Apr 13 '14 at 14:37

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