Algebraic number

In mathematics , an algebraic number is a real or complex number, the root of a polynomial of degree greater than zero (non-constant polynomial) ${\displaystyle x}$

${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{1}x+a_{0}}$

with rational coefficients , i.e. solution of the equation . ${\displaystyle a_{k}\in \mathbb {Q} ,k=0,\dotsc ,n,a_{n}\neq 0}$ ${\displaystyle f(x)=0}$

The algebraic numbers defined in this way form a real subset of the complex numbers . Apparently every rational number is algebraic because it solves the equation . So it applies . ${\displaystyle \mathbb {A} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle q}$ ${\displaystyle x-q=0}$ ${\displaystyle \mathbb {Q} \subsetneq \mathbb {A} \subsetneq \mathbb {C} }$

If a real (or more generally complex) number is not algebraic, it is called transcendent .

The also common definition of algebraic numbers as zeros of polynomials with integer coefficients is equivalent to the one given above. Any polynomial with rational coefficients can be converted into one with integer coefficients by multiplying by the main denominator of the coefficients. The resulting polynomial has the same zeros as the starting polynomial.

Polynomials with rational coefficients can be normalized by dividing all coefficients by the coefficient . Zeros of normalized polynomials whose coefficients are integers, is called ganzalgebraische numbers or all algebraic numbers. The whole algebraic numbers form a subring of the algebraic numbers, but which is not factorial . For the general concept of wholeness, see wholeness (commutative algebra) . ${\displaystyle a_{n}}$

The concept of the algebraic number can be extended to that of the algebraic element by taking the coefficients of the polynomial from an arbitrary field instead of from . ${\displaystyle \mathbb {Q} }$

Degree and minimal polynomial of an algebraic number

For many investigations of algebraic numbers, the degree defined below and the minimal polynomial of an algebraic number are important.

Is an algebraic number that is an algebraic equation ${\displaystyle x}$

${\displaystyle f(x)=x^{n}+\dotsb +a_{1}x+a_{0}=0}$

with , fulfilled, but no such equation of a lesser degree, then one calls the degree of . Thus all rational numbers are of degree 1. All irrational square roots of rational numbers are of degree 2. ${\displaystyle n\geq 1}$ ${\displaystyle a_{k}\in \mathbb {Q} }$ ${\displaystyle n}$ ${\displaystyle x}$

The number is also the degree of the polynomial , the so-called minimal polynomial of . ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle x}$

Examples

• For example, is an integer algebraic number because it is a solution to the equation . Likewise, the imaginary unit as a solution to is entirely algebraic. ${\displaystyle {\sqrt {2}}}$ ${\displaystyle x^{2}-2=0}$ ${\displaystyle i}$ ${\displaystyle x^{2}+1=0}$
• ${\displaystyle {\sqrt {2}}+{\sqrt {3}}}$is a whole algebraic number of degree 4. See example for algebraic element .
• ${\displaystyle {\tfrac {1}{2}}}$and are examples of algebraic numbers 1st and 2nd degree, which are not entirely algebraic. ${\displaystyle {\tfrac {1}{\sqrt {2}}}}$
• Towards the end of the 19th century it was proven that the circle number and Euler's number are not algebraic. Of other numbers, such as, for example , it is still not known whether they are algebraic or transcendent. See the article Transcendent Number . ${\displaystyle \pi }$ ${\displaystyle e}$ ${\displaystyle \pi +e}$

properties

The set of algebraic numbers is countable and forms a field.

The field of algebraic numbers is algebraically closed ; that is, every polynomial with algebraic coefficients has only algebraic zeros. This body is a minimal algebraically closed upper body of and is therefore an algebraic closure of . One often writes it as (for "algebraic concluding "; interchangeable with other final terms) or (for " A lgebraische numbers"). ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle {\overline {\mathbb {Q} }}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {A} }$

Above the field of rational numbers and below the field of algebraic numbers there are an infinite number of intermediate fields; roughly the set of all numbers of the form , where and are rational numbers and is the square root of a rational number . The body of the points on the complex plane of numbers that can be constructed with compasses and ruler is also such an algebraic intermediate body. ${\displaystyle a+b\cdot q}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle q}$ ${\displaystyle r}$ ${\displaystyle \{0,1\}}$

See also : Euclidean body

In the context of Galois theory , these intermediate bodies are examined in order to gain deep insights into the solvability or unsolvability of equations. One result of Galois theory is that every complex number that can be obtained from rational numbers by using the basic arithmetic operations ( addition , subtraction , multiplication and division ) and by taking n -th roots ( n is a natural number) (these are called numbers "Can be represented by radicals"), is algebraic, but conversely there are algebraic numbers that cannot be represented in this way; all these numbers are zeros of polynomials at least 5th degree.

Web links

• Barry Mazur : Algebraic Numbers. (PDF; 272 kB).

Individual evidence