<s>[INST] Factoring irreducible fractions Is there a technique or a formula that factors irreducible fractions into products of polynomials, for example:
$$\frac1{x^8+1}=\frac{1}{(x^4+ \sqrt2x^2+1)(x^4- \sqrt2x^2+1)}$$
Also, is there a way to factor, in the same way as above, the following fraction:
$$\frac1{x^4-x^2+1}$$
I am not interested in the completition of the square. Thank you.
 [/INST] One has a direct way, writing
$$
x^8+1=(x^8+2x^4+1)-2x^4=(x^4+1)^2-2x^4=(x^4+ \sqrt2x^2+1)(x^4- \sqrt2x^2+1)
$$ similarly
$$
x^4-x^2+1=(x^4+1)-x^2=(x^2+1)^2-3x^2=(x^2+ \sqrt3x+1)(x^2- \sqrt3x+1).
$$
</s>