Find the greatest common divisor of the polynomials $ f(x)=x^{4}+ix^{2}+1 $ and $g(x)=ix^{2}+1 $ over $\mathbb{C} $

Question

Find the greatest common divisor of the polynomials $f(x)=x^{4}+ix^{2}+1$ and $g(x)=ix^{2}+1$ over the field $\mathbb{C}$.

Here I am getting some difficulty because I think both are prime and hence $\gcd =1$ , but I am not sure. Any help

Answer 1

The gcd= 1.

$x^4+ix^2+1=(ix^2+1)(\dfrac{x^2}{i}+2) - 1$

Answer 2

First note that $$\gcd(x^4+ix^2+1,ix^2+1)=\gcd(x^4,ix^2+1)$$ and that $x^4$ and $ix^2 + 1$ don't have a common linear factor since they don't have common zeroes. All irreducible polynomials over $\Bbb C$ are linear, so those polynomials don't have common irreducible factors. We conclude that GCD is $1$.

Note. This a bit lengthy explanation, but easier than long division. Alternatively, note that $$1+x^4 = (1+ix^2)(1-ix^2).$$