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Are there $a(x),b(x) \in \mathbb{Q}[x]$ such that $a(x)f(x)+b(x)g(x) = x^2 + 1$

Where $f(x) = x^4+4x^3-7x+2$ and $g(x)=x^2+3x-4$

I have no idea how to approach this problem. I tried to visualize it as one with integers, but there seems to be some missing info - one function with two unknowns. Is there a special property of polynomials I'm missing here?

Alex Osheter
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  • This is probably intended as a ring theory problem. The question is if $x^2+1$ is in the ideal $(f(x),g(x))$. – pancini Apr 12 '20 at 09:23
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    Since you have gone and visualized it as integers, doesn't it remind you of Bezout, or the euclidean method for the gcd? – Exodd Apr 12 '20 at 09:23
  • @Exodd I'm not aware of Bezout, but definitely did not see the resemblance to the euclidean method. – Alex Osheter Apr 12 '20 at 09:29
  • It's the same as for the integers, using the Bezout identity for the GCD - see my answer in the linked dupe.The same method works in any Bezout domain, i.e where the gcd may be expressed as a linear combination of its args, i.e. where finitely generated ideals are principal. – Bill Dubuque Apr 12 '20 at 21:17

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Well, for any polynomials $f(x),g(x)$ with coefficients in a field like the rational numbers, one can compute their greatest common divisor (gcd) $d(x)$ and represent it as a linear combination of these polynomials by using the extended Euclidean algorithm, i.e.

$f(x)s(x) + g(x)t(x) = d(x).$

In your example, you have to check if the right-hand side is the gcd or a multiple of it (and so lies in the ideal generated by $f(x)$ and $g(x)$), or it is not in the ideal (then the equation has no solutions $s(x)$ and $t(x)$).

Wuestenfux
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    Wouldn't it be possible to achieve $d(x)$ without it being a gcd or a multiple thereof? Since we are dealing with addition here. To bring it back to integers for example: 14 + 17 = 11. But 11 is not the gcd of 4 nor 7. – Alex Osheter Apr 12 '20 at 09:32
  • That's why he wrote "or a multiple of it". The ideal equality $, f R + g R = d R, $ means the representable elemnts are precisely the multiples of $d$ in $R = \Bbb Q[x]\ $ – Bill Dubuque Apr 12 '20 at 21:21