Chapter 11 (Rings) – Artin – Section 1 (Definition of a Ring)

1.

Note that a complex number \alpha is algebraic if it is a root of a(nonzero) polynomial with integer coefficients.
a) Sine we need 7 + \sqrt [3]{2} to be root of an integer polynomial, there fore (x - (7 + \sqrt [3]{2}) = (x-7 - 2^{\frac{1}{3}}) is one of the factors. Clearly (x-7)^3 - (\sqrt [3]{2})^3 = (x-7)^3 - 2 is the desired polynomial.
b) Again we need \sqrt 3 + \sqrt {-5} as a root of an integer polynomial. Therefore (x - (\sqrt 3 + \sqrt {-5})) is one of the factors. Take another factor as (x - (\sqrt 3 - \sqrt {-5})) (motivation is to get rid of square roots).
(x - (\sqrt 3 + \sqrt {-5})) \times (x - (\sqrt 3 - \sqrt {-5})) = x^2 - 2\sqrt 3 + 8
Finally we multiply x^2 + 8 + 2\sqrt 3 to that to get rid of the final square root.
(x^2 - 2\sqrt 3 + 8) \times (x^2 + 8 + 2\sqrt 3) = (x^2 + 8)^2 - 12 x^2

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