Algebraic number: Difference between revisions
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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, and every complex root of a polynomial with integer coefficients is an algebraic number. | 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, and every complex root of a polynomial with integer coefficients is an algebraic number. If an algebraic number 'x' can be written as the root of a [[monic polynomial]], | ||
that is, one whose [[leading coefficient]] is 1, then 'x' is called an ''algebraic integer''. | |||
The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are [[countable set|countable]]. The algebraic numbers form a [[field (mathematics)|field]]; in fact, they are the smallest [[algebraically closed field]] with characteristic 0. <ref>If 1 + 1 = 0 in the field, the characteristic is said to be 2; if 1 + 1 + 1 = 0 the characteristic is said to be 3, and forth. If there is no <math>n</math> such that adding 1 <math>n</math> times gives 0, we say the characteristic is 0. A field of positive characteristic need not be finite. </ref> | The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are [[countable set|countable]]. The algebraic numbers form a [[field (mathematics)|field]]; in fact, they are the smallest [[algebraically closed field]] with characteristic 0. <ref>If 1 + 1 = 0 in the field, the characteristic is said to be 2; if 1 + 1 + 1 = 0 the characteristic is said to be 3, and forth. If there is no <math>n</math> such that adding 1 <math>n</math> times gives 0, we say the characteristic is 0. A field of positive characteristic need not be finite. </ref> | ||
Revision as of 13:51, 8 November 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, and every complex root of a polynomial with integer coefficients is an algebraic number. If an algebraic number 'x' can be written as the root of a monic polynomial, that is, one whose leading coefficient is 1, then 'x' is called an algebraic integer.
The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are countable. The algebraic numbers form a field; in fact, they are the smallest algebraically closed field with characteristic 0. [1]
Real or complex numbers that are not algebraic are called transcendental numbers.
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 .
Notes
- ↑ If 1 + 1 = 0 in the field, the characteristic is said to be 2; if 1 + 1 + 1 = 0 the characteristic is said to be 3, and forth. If there is no such that adding 1 times gives 0, we say the characteristic is 0. A field of positive characteristic need not be finite.
