Quadratic irrational

In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the set of rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their common denominator, a quadratic irrational is an irrational root of some quadratic equation whose coefficients are integers. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as

for integers $a, b, c, d$ ; with $b$ , $c$ and $d$ non-zero, and with $c$ square-free. When $c$ is positive, we get real quadratic irrational numbers, while a negative $c$ gives complex quadratic irrational numbers which are not real numbers. This implies that the quadratic irrationals have the same cardinality as ordered quadruples of integers, and are therefore countable.

Quadratic irrationals are used in field theory to construct field extensions of the rational field $ℚ$ . Given the square-free integer $c$ , the augmentation of $ℚ$ by quadratic irrationals using $\sqrt c$ produces a quadratic field $ℚ(\sqrt c$ ). For example, the inverses of elements of $ℚ(\sqrt c$ ) are of the same form as the above algebraic numbers:

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