Algebraic number: Difference between revisions

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imported>Wlodzimierz Holsztynski
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A momentary reflection gives now
A momentary reflection gives now


'''Theorem''' The degree of the inverse of any algebraic number </math>\ z\ne 0</math>&nbsp; is equal to the degree of the number </math>\ z</math>&nbsp; itself.
'''Theorem''' The degree of the inverse of any algebraic number <math>\ z\ne 0</math>&nbsp; is equal to the degree of the number <math>\ z</math>&nbsp; itself.


== The sum and product of two algebraic numbers ==
== The sum and product of two algebraic numbers ==

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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. 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, and, in fact, an algebraic integer, as it is a root of the polynomial . Similarly, the imaginary unit is an algebraic integer, being a root of the polynomial .

Algebraic numbers via subalgebras and subfields

The field of complex numbers   is a linear space over the field of rational numbers   In this section, by a linear space we will mean a linear subspace of   over   and by algebra we mean a linear space which is closed under the multiplication, and which has   as its element. The following properties of a complex number   are equivalent:

  •   is an algebraic number of degree
  •   belongs to an algebra of linear dimension

Indeed, when the first condition holds, then the powers   linearly generate the algebra required by the second condition. And if the second condition holds then the   elements 1,z,\dots,z^{n}</math>  are linearly dependent (over rationals).

Actually, every finite dimensional algebra   is a field—indeed, divide an equality

where   by   and you quickly get an equality of the form:

A momentary reflection gives now

Theorem The degree of the inverse of any algebraic number   is equal to the degree of the number   itself.

The sum and product of two algebraic numbers

Let   and   where   are finite linear bases of fields   respectively. Let   be the smallest algebra generated by   Then   is linearly generated by

Thus the linear dimensions (over rationals) of the three algebras satisfy inequality:

Now, let   be arbitrary algebraic numbers of degrees   respectively. They belong to their respective m- and n-dimensional algebras. The sum and product   belong to the algebra generated by the union of the two mentioned algebras. The dimension of the generated algebra is not greater than It contains   as well as all linear combinations   with rational coefficients   This proves:

Theorem  The sum and the product of two algebraic numbers of degree m and n, respectively, are algebraic numbers of degree not greater than mn. The same holds for the linear combinations with rational coefficients of two algebraic numbers.

As a corollary to the above theorem, together with the previous section, we obtain:

Theorem  The algebraic numbers form a field.

Notes

  1. 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.