The difference between Algebraic integer and Algebraic number

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 algebraic number means a complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers.

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

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

    A complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers.

    Examples:

    "The golden ratio (&phi;) is an [[algebraic number]] since it is a solution of the quadratic equation <math> x^2 + x - 1 = 0 </math>, whose coefficients are integers."

    "The square root of a rational number, <math>\textstyle\sqrt{\frac m n},</math> is an [[algebraic number]] since it is a solution of the quadratic equation <math>n x^2 - m = 0</math>, whose coefficients are integers."

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