Results 1 to 5 of 5

Thread: How to use Euclidian Algorithm to find multiplicative inverse of a polynomial??

  1. #1
    Member elninio's Avatar
    Joined
    Sep 2009
    Posts
    92
    Awards
    1

    How to use Euclidian Algorithm to find multiplicative inverse of a polynomial??

    For example, finding the multiplicative inverse of$\displaystyle [x] in Z_5[x]/<x^2+x+1>$

    or even something like

    $\displaystyle [a+bx] in R[x]/<x^2+1>$

    I cant seem to figure out this concept.

    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by elninio View Post
    For example, finding the multiplicative inverse of$\displaystyle [x] in Z_5[x]/<x^2+x+1>$

    or even something like

    $\displaystyle [a+bx] in R[x]/<x^2+1>$

    I cant seem to figure out this concept.

    Thank you!

    Divide $\displaystyle x^2+1$ by $\displaystyle bx+a$ with residue:

    $\displaystyle x^2+1=(bx+a)(dx+c)+r$ , with $\displaystyle r=0\,\,\,or\,\,\,\deg(r)<1\Longrightarrow r\in\mathbb{R}\setminus{0}$.

    As $\displaystyle x^2+1$ has no real roots, it can't be $\displaystyle r=0\Longrightarrow 0\ne r\in\mathbb{R}$ , and thus $\displaystyle (bx+a)\left(\frac{d}{r}x+\frac{c}{r}\right)=1\!\!\ !\pmod{x^2+1}\Longrightarrow (bx+a)^{-1}=\frac{d}{r}x+\frac{c}{r}\,\,\,in\,\,\,\mathbb{R }[x]\slash<x^2+1>$

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    Tonio, I think, the inverse of bx+a should be -(dx+c)/r.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by Shanks View Post
    Tonio, I think, the inverse of bx+a should be -(dx+c)/r.

    Yes, of course: forgot the sign. Thanx

    Tonio
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Jul 2009
    Posts
    193
    Thanks
    5
    Can we not write it in terms of a and b like

    $\displaystyle (bx+a)^{-1}=\frac{a}{a^2+b^2}-\frac{b}{a^2+b^2}x$?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Extended Euclid's algorithm - multiplicative inverse
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Nov 12th 2010, 01:45 AM
  2. How to find inverse of polynomial equation?
    Posted in the Pre-Calculus Forum
    Replies: 11
    Last Post: Apr 28th 2010, 08:03 PM
  3. Multiplicative Inverse
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Dec 4th 2009, 06:47 AM
  4. Multiplicative inverse 71.7
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Oct 26th 2009, 09:34 AM
  5. Multiplicative inverse
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 14th 2006, 01:53 PM

Search tags for this page

Search Tags


/mathhelpforum @mathhelpforum