Algebraic number: Difference between revisions
imported>Martin Goldstern (transcendental. x^2-2) |
imported>Michael Hardy ("can be complex" is an understatement: all algebraic numbers are complex.) |
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An '''algebraic number''' is a root of a [[polynomial]] with rational coefficients. | An '''algebraic number''' is any [[complex number]] that is a root of a [[polynomial]] with rational coefficients. Any polynomial with rational coefficients can be converted to one with integer coefficients by multiplying through by the least common multiple of the denominators, so an algebraic number is also a root of a polynomial with integer coefficients. | ||
The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are [[countable set|countable]]. The algebraic numbers for a field; in fact, they are the smallest [[algebraically closed field]] with characteristic 0. | The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are [[countable set|countable]]. The algebraic numbers for a field; in fact, they are the smallest [[algebraically closed field]] with characteristic 0. | ||
Revision as of 16:35, 28 April 2007
An algebraic number is any complex number that is a root of a polynomial with rational coefficients. Any polynomial with rational coefficients can be converted to one with integer coefficients by multiplying through by the least common multiple of the denominators, so an algebraic number is also a root of a polynomial with integer coefficients.
The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are countable. The algebraic numbers for a field; in fact, they are the smallest algebraically closed field with characteristic 0.
Real or complex numbers that are not algebraic are called transcendental numberss.
Examples
is an algebraic number, as it is a root of the polynomial . Similarly, the imaginary unit is algebraic, being a root of the polynomial .
