theory of algebraic and transcendental numbers

theory of algebraic and transcendental numbers

The following entry is some sort of index of articles in PlanetMath about the basic

theory of algebraic and transcendental numbers

, and it should be studied together with its

complement

: the

theory of rational and irrational numbers

. The reader should follow the links in each bullet-point to learn more about each topic. For a somewhat deeper approach to the subject, the reader should read about

Algebraic Number Theory

. In this entry we will concentrate on the properties of the

complex numbers

and the

extension

ℂ

/

ℚ

, however, in general, one can talk about numbers of any field

F

which are algebraic over a

subfield

K

.

1

Basic Definitions

1.

A number

α

∈

ℂ

is said to be

algebraic

(

http://planetmath.org/Algebraic

) (over

ℚ

), or an

algebraic number

, if there is a

polynomial

p

⁢

(

x

)

with

integer

coefficients such that

α

is a root of

p

⁢

(

x

)

(i.e.

p

⁢

(

α

)

=

0

).

2.

Similarly as the

rational numbers

may be classified to integer and non-integer (fractional) numbers, also the algebraic numbers may be classified to

algebraic integers

or algebraic numbers and non-integer algebraic numbers.  The algebraic integers form an

integral domain

.

3.

The numbers

-

12

,

2

,

7

3

,

2

+

7

3

,

ζ

7

=

e

2

⁢

π

⁢

i

/

7

(that is, a

7

th

root of unity

), are all algebraic integers,

2

2

is a non-integer algebraic number (its is

2

⁢

x

2

-

1

).  See also

rational algebraic integers

.

4.

A number

α

∈

ℂ

is said to be transcendental if it is not algebraic.

5.

For example, e is transcendental, where e is the natural

log

base (also called the

Euler number

). The number Pi (

π

) is also transcendental. The proofs of these two facts are HARD!

6.

A field extension

L

/

K

is said to be an

algebraic extension

if every element of

L

is algebraic over

K

. An extension which is not algebraic is said to be transcendental. For example

ℚ

⁢

(

2

)

/

ℚ

is algebraic while

ℚ

⁢

(

e

)

/

ℚ

is transcendental (see the

simple field extensions

).

7.

The

algebraic closure

of a field

ℚ

is the union of all algebraic extension fields

L

of

ℚ

. The algebraic closure of

ℚ

is usually denoted by

ℚ

¯

. In other words,

ℚ

¯

is the union of all complex numbers which are algebraic.

8.

The set

ℚ

¯

of all algebraic numbers is a field.  It has as a subfield the

ℚ

¯

∩

ℝ

, the set of all real algebraic numbers, and as a

subring

the set of all algebraic integers.  See the

field of algebraic numbers

and the ring of algebraic integers.

9.

The ring of all algebraic integers

𝔸

contains no

irreducible elements

(

http://planetmath.org/RingWithoutIrreducibles

).

10.

The height of an algebraic number is a way to measure the complexity of the number.

2

Small Results

1.

A

finite extension

of fields is an algebraic extension.

2.

The extension

ℝ

/

ℚ

is not finite (

http://planetmath.org/ExtensionMathbbRmathbbQIsNotFinite

).

3.

For every algebraic number

α

, there exists an irreducible

minimal polynomial

m

α

⁢

(

x

)

such that

m

α

⁢

(

α

)

=

0

(see

existence

of the minimal polynomial).

4.

For any algebraic number

α

, there is a nonzero

multiple

n

⁢

α

which is an algebraic integer (see multiples of an algebraic number):

5.

Some examples of algebraic numbers are the sine, cosine and

tangent

of the angles

r

⁢

π

where

r

is a rational number (see this entry (

http://planetmath.org/AlgebraicSinesAndCosines

)).  More usual are the root expressions of rational numbers.

6.

The

transcendental root theorem

(

http://planetmath.org/ProofOfTranscendentalRootTheorem

): Let

F

⊂

K

be a field extension with

K

an algebraically closed field. Let

x

∈

K

be transcendental over

F

. Then for any

natural number

n

≥

1

, the

element

x

1

/

n

∈

K

is also transcendental over

F

.

7.

An

example of transcendental number

(as an application of Liouville’s

approximation theorem

).

8.

The

algebraic numbers are countable

. In other words,

ℚ

¯

is a

countable

subset of

ℂ

. Since

ℂ

is

uncountable

, we conclude that there are infinitely many

transcendental numbers

(uncountably many!).  See also the proof of the existence of transcendental numbers.

9.

Algebraic and transcendental:  the sum,

difference

, and

quotient

of two non-zero complex numbers, from which one is algebraic and the other transcendental, is transcendental.

10.

All

transcendental extension

fields

ℚ

⁢

(

α

)

of

ℚ

are

isomorphic

(see the

simple transcendental field extensions

).

3

BIG Results

1.

Steinitz Theorem

: There exists an algebraic closure of a field.

2.

The

Gelfond-Schneider Theorem

: Let

α

and

β

be algebraic over

ℚ

, with

β

irrational

and

α

not equal to 0 or 1. Then

α

β

is transcendental over

ℚ

.

3.

The

Lindemann-Weierstrass Theorem

.

Title

theory of algebraic and transcendental numbers

Canonical name

TheoryOfAlgebraicAndTranscendentalNumbers

Date of creation

2013-03-22 15:14:01

Last modified on

2013-03-22 15:14:01

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

32

Author

alozano (2414)

Entry type

Topic

Classification

msc 11R04

Related topic

AlgebraicNumberTheory

Related topic

TheoryOfRationalAndIrrationalNumbers

Related topic

MultiplesOfAnAlgebraicNumber

Related topic

NormAndTraceOfAlgebraicNumber

Related topic

AlgebraicSumAndProduct

Related topic

DegreeOfAnAlgebraicNumber

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