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The question is to find the greatest common divisor (in the form of monic polynomial) in $\Bbb F_5[X]$ of

$f=x^2-x+4$ and $g=x^3+2x^2+3x+2$

I used the Euclidean Algorithm for polynomials and found that the the GCD of these two polynomials is $2$.

How do I make this a monic polynomial?

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2 Answers 2

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In general rings or domains, gcds are defined only up to unit factors, since if $\,c\mid a,b\,$ then so too does $\,uc\,$ for $\,u\,$ any unit (invertible). In $\,\Bbb Z\,$ we can normalize gcds by choosing a nonnegative rep, i.e. multiply the gcd by $\,u=-1\,$ to make it positive if need be. Similarly, in your example, for polynomials over a field, we may normalize gcds by scaling them to be monic, i.e. scale the polynomial by the inverse of its leading coefficient to force the lead coefficient to be $\,1$.

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  • $\begingroup$ Therefore my answer would be 2/2 = 1? $\endgroup$
    – jsan
    Commented Mar 20, 2014 at 16:58
  • $\begingroup$ @jsan Yes, or equivalently, multiply by $\ \dfrac{1}2 \equiv \dfrac{6}2\equiv 3\pmod 5\ \ $ $\endgroup$ Commented Mar 20, 2014 at 17:23
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What do you get if you multiply your GCD by 3?

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