Discussion
Let \(\mathbf{R}[X]\) be the polynomial ring in one indeterminate over the field of real numbers. Then in the quotient ring \(\mathbf{R}[X]/\lt X^{8} + 1\gt\) number of maximal ideals is :
Answer: D
The quotient has 4 maximal ideals.
The roots of \(X^{8} + 1\) are the eight primitive 16th roots of unity, none of them real. They form four conjugate pairs, so over \(\mathbf{R}\) the polynomial factors into four distinct irreducible quadratics \(q_{1}q_{2}q_{3}q_{4}\).
Maximal ideals of \(\mathbf{R}[X]/\langle X^{8} + 1 \rangle\) correspond to maximal ideals of \(\mathbf{R}[X]\) containing \(X^{8} + 1\), that is, to its distinct irreducible factors. By the Chinese remainder theorem the ring is \(\mathbf{C}^{4}\), with exactly 4 maximal ideals.
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