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Use the Euclidean Algorithm to find the GCD of $3x^2+1$ and $x+1$ in $\mathbb{Z}_5[x]$.

What I got was this:

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

$x+1 = (4)(4x+4) + 0$

Since 4 was the last non-zero remainder, that is our GCD for the two polynomials. But, what does this mean?

mathnoob
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    You can apply the remainder theorem to conclude that $4 = 3 \cdot (-1)^2 + 1$. – Mark Saving May 17 '22 at 01:21
  • Oh! So, then it is remainder 1 and thus the two polynomials are prime? – mathnoob May 17 '22 at 01:22
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    yeah since any non zero constant is a unit you have that their $gcd$ is $1$ so they are relatively prime – MIO May 17 '22 at 01:46
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    $4=-1$ is a unit (invertible) in $\Bbb Z_5$ so the gcd (unit) normalizes to $1,$ see here in teh linked dupe. – Bill Dubuque May 17 '22 at 01:59
  • thank you all for the clarification! – mathnoob May 17 '22 at 02:03
  • @mathnoob The remainder is indeed 4, not 1. You calculated it correctly. If you wished to compute the gcd, you would continue with the Euclidean algorithm by taking the remainder of $x + 1$ divided by $4$, which would give you a remainder of zero. This would enable you to conclude that $4$ is the gcd, which is equivalent to saying 1 is the gcd since the gcd is defined only up to multiplication with a unit. – Mark Saving May 17 '22 at 02:05

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