The difference between Algebraic integer and Quadratic integer

When used as nouns, algebraic integer means a real or complex number (more generally, an element of a number field) which is a root of a monic polynomial whose coefficients are integers, whereas quadratic integer means a number (real or complex) which is a solution to an equation of the form x^2 + bx + c = 0 for some integers b and c. for example 42, or 2+3i, or 1+\sqrt{-3} \over 2.

check bellow for the other definitions of Algebraic integer and Quadratic integer

  1. Algebraic integer as a noun (algebra, number theory):

    A real or complex number (more generally, an element of a number field) which is a root of a monic polynomial whose coefficients are integers; equivalently, an algebraic number whose minimal polynomial (lowest-degree polynomial of which it is a root and whose leading coefficient is 1) has integer coefficients.

    Examples:

    "A Gaussian integer <math> z = a + i b </math> is an [[algebraic integer]] since it is a solution of either the equation <math> z^2 + (-2 a) z + (a^2 + b^2) = 0 </math> or the equation <math> z - a = 0 </math>."

  1. Quadratic integer as a noun (mathematics):

    A number (real or complex) which is a solution to an equation of the form x^2 + Bx + C = 0 for some integers B and C. For example 42, or 2+3i, or 1+\sqrt{-3} \over 2.

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