Definition An element \( \alpha \) of \( \mathbb C \) is an algebraic integer of degree two (alternatively, a quadratic algebraic integer) if there exists a polynomial of the form \( P (X ) = X^ 2 + aX + b \) with a, b integers roots such that \( P (X ) \) has no rational roots and \( P (\alpha) = 0 \).
Proposition If \( \alpha \) is an algebraic integer of degree two, then \( \mathbb Z [\alpha] \) is equal to the set of complex numbers of the form \( x + y\alpha \), where x and y are integers.
Proof. Let set S be the set of complex numbers of the form \( x + y\alpha \), where x and y are integers.
Then it is obvious that S is a subset of \( \mathbb Z [\alpha] \).
\( \forall x,y,z \in S, xy=x_2 y_2 \alpha^2 + (x_1y_2+x_2y_1) \alpha + x_1 y_1= (x_2 y_2)\cdot(-a\alpha-b)+ (x_1y_2+x_2y_1) \alpha + x_1 y_1 = \)\( yx \implies xy \in S \). Note \( (x+y)z=xz+yz \). So S is a subring of \( \mathbb C \). \( \alpha \in S \implies S \supseteq \mathbb Z [\alpha] \)
\( S \subseteq \mathbb Z [\alpha] \, \land \, S \supseteq \mathbb Z [\alpha] \implies S=\mathbb Z [\alpha] \)
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