Factor x⁴+1

Completely factor p(x) = x^4 + 1.


Evidently, p(x) = (x - \frac{\sqrt{2}}{2}(1+i))(x + \frac{\sqrt{2}}{2}(1+i))(x - \frac{\sqrt{2}}{2}(1-i))(x + \frac{\sqrt{2}}{2}(1-i)). (WolframAlpha agrees.)

To find this factorization, note that x^4 + 1 = (x^2 + i)(x^2 - i) = (x + \sqrt{i})(x - \sqrt{i})(x + \sqrt{-i})(x - \sqrt{-i}). Next, suppose (for instance) that (a+bi)^2 = i; comparing coefficients, we see that a = b = \sqrt{2}/2 is one solution. The other roots can be simplified in a similar way.

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