Finding the $\gcd$ of $a(x)$ and $b(x)$ in field $\mathbb{F}$

Question

I'm trying to find the $\gcd$ of $a(x) = x^4 + 2x^3+x^2+4x+2$ and $b(x)=x^2+3x+1$ over $\mathbb{F_5}$. I've already tried Euclid's algorithm:

$x^4 + 2x^3+x^2+4x+2 = x^2(x^2+3x+1) - x^3+4x+2$. Now I should express $b(x)$ in terms of the remainder $-x^3+4x+2$, but I'm not sure how to do this since $\deg(b(x)) < 3$. Did I do something incorrectly? How do I find the $\gcd$?

You didn't complete the (Euclidean) division - doing so yields a remainder of smaller degree than the divisor. — Bill Dubuque, Jan 25 '19 at 18:00

score 1 · Answer 1 · answered Jan 25 '19 at 17:19

1

It is perhaps easier just to factorize the polynomials over $\Bbb F_5$ instead of using the Euclidean algorithm:

$$ x^4+2x^3+x^2+4x+2=(x^3+3x^2+4x+3)(x+4) $$

$$ x^2+3x+1=(x+4)^2 $$

For this is enough to look for roots. The second polynomial obviously has $x=-4=1$ as a double root. So what about the first polynomial? Clearly it has also a root $1$. What about the cubic polynomial left?

So you see that $gcd(a(x),b(x))=x+4$ over $\Bbb F_5$. Note that the gcd is $1$ over $\Bbb Z$.

answered Jan 25 '19 at 17:19

Dietrich Burde

130,978

Thanks! One more question: assume I want to find the coefficients $\lambda (x), \mu (x)$ such that $\gcd(a(x),b(x)) = \lambda (x) a(x) + \mu (x) b(x)$. Can I find them using your method? – MyWorld Jan 25 '19 at 17:55
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@Zachary Use the Extended Euclidean algorithm, e.g. see here. – Bill Dubuque Jan 25 '19 at 18:03

Finding the $\gcd$ of $a(x)$ and $b(x)$ in field $\mathbb{F}$

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