Algebraic independence

In Algèbre, the algebraic independence of a Ensemble on a body describes the fact that its elements are not roots of a Polynôme with coefficients in this body.

Definition

That is to say

L

a body,

S

a Subset of

L

and

K

a subfield of

L

S

is algebraically independent on

K

if the elements of

S

are not roots of any noncommonplace Polynôme with coefficients in

L

In other words, for any finished continuation $(\ alpha_1, \ ldots, \ alpha_n)$ of elements distinct from $S$ and any not-commonplace polynomial $P (x_1, \ ldots, x_n)$ with coefficients in $K$ :

P (\ alpha_1, \ dowries, \ alpha_n) \ 0

In particular, a whole with only one element $\ {\ alpha \}$ is algebraically independent on $K$ if and only if $\ alpha$ is transcendent on $K$ .

Examples

The subset

\ {\ sqrt {\ pi}; 2 \ pi +1 \}

of the body of the real numbers

\ mathbb R

is not algebraically independent of the body of the rational numbers

\ mathbb Q

since the polynomial

P (x_1, x_2) =2x^2_1-x_2+1

is not commonplace and with coefficients in

\ mathbb Q

and

P (\ sqrt {\ pi}, 2 \ pi +1) =0

The Théorème of Lindemann-Weierstrass can often be used to prove that certain units are algebraically independent on $\ mathbb Q$ .

It is not known if the unit $\ {\ pi, E \}$ is algebraically independent on $\ mathbb Q$ . Yu Nesterenko proved into 1996 that $\ {\ pi, e^ \ pi, \ Gamma (1/4) \}$ is.

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