TIFR-2012-Maths-D: 32

Question

True/False Question:

The polynomial $X^{8}+1$ is irreducible in $\mathbb{R}\left [ X \right ]$.

Answer 1

$x^{8}+1=(x^{4}+1)^2 -2x^4=(x^{4}+1)^2 -(\sqrt{2} x)^2 =(x^{4}+1+ \sqrt{2} x)(x^{4}+1-\sqrt{2} x)$

The field of complex numbers is Algebraically Closed and it is the degree 2 extension of Real field. Hence any polynomial of degree >2 in R[X] is reducible.