MHB Why is x^2 + 1 an Irreducible Polynomial?

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Why is x^2 + 1 irreducible?
 
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RTCNTC said:
Why is x^2 + 1 irreducible?

What do you think? If you factor this expression, what do you get? Moreover, what could be said about those factors in relation to the real numbers?
 
Complex numbers?
 
Indeed!

In this case we could say that the expression is irreducible over the reals, but reducible over the complex numbers. But I might let someone else chime into give a more formal approach. :)
 
Cool.
 
What about x^2 + x + 1?
 
RTCNTC said:
What about x^2 + x + 1?

What does the discriminant tell you about the roots of this quadratic polynomial?
 
b^2 - 4ac

1^2 - 4(1)(1)

1 - 4

-3

b^2 - 4ac < 0

This means no real solutions.
 

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