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Show that x^2 + x + 1 is prime.
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Consider the discriminant of this quadratic polynomial.
The discriminant is
d = b^2 - 4ac = 1^2 - 4*1*1 = -3.
The discriminant is negative, which means that the polynomial does not have real roots.
If this polynomial be a composite over real numbers, it would be a product of two linear
binomials (ax+b)*(cx+d).
But then it would have two real roots, -b/a and -c/d.
Thus we get a CONTRADICTION with the fact proven above that this polynomial has no real roots.
The contradiction PROVES that the polynomial is not a composite over real numbers.
Hence, it is a PRIME over real numbers.
Solved.
