proof of algebraic independence of elementary symmetric polynomials
Geometric proof, works when R is a division ring.
Consider the quotient field Q of R and then the algebraic closure of .
Consider the substitution map that associates to values the symmetric functions in these variables .
And by developing the right-hand side we get .
The capital letters are there to emphasize the and are variables and and are regarded as function algebras over .
The theorem stating that the symmetric functions are algebraically independent is no more than saying that this morphism is injective.
As a matter of fact, is the symmetric function in the , and is clearly a morphism of algebras.
The conclusion is then straightforward from the surjectivity of because if for some , then by surjectivity of it means that was zero in the first place. In other words the kernel of is reduced to 0.
|Title||proof of algebraic independence of elementary symmetric polynomials|
|Date of creation||2013-03-22 17:38:28|
|Last modified on||2013-03-22 17:38:28|
|Last modified by||lalberti (18937)|