|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
proof of algebraic independence of elementary symmetric polynomials
|
(Proof)
|
|
|
Geometric proof, works when R is a division ring.
Consider the quotient field Q of R and then the algebraic closure $K$ of $Q$ .
Consider the substitution map that associates to values $t_1,\ldots,t_n \in K^n$ the symmetric functions in these variables $s_1,\ldots,s_n$ .
Because $K$ is algebraic closed this map is surjective. Indeed, fix values $v_i$ , then on an algebraic closed field there are roots $t_i$ such that $$X^n + \sum_i v_i X^i = \Pi_i (X+t_i)$$ And by developing the right-hand side we get $v_i = s_i$ .
Then we consider the transposition morphism of algebras $\phi^*$ :
The capital letters are there to emphasize the $S_i$ and $T_i$ are variables and $R[S_1,\ldots,S_n]$ and $R[T_1,\ldots,T_n]$ are regarded as function algebras over $K^n$ .
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, $\phi^*(S_i)$ is the $i^{th}$ symmetric function in the $T_i$ , and $\phi^*$ is clearly a morphism of algebras.
The conclusion is then straightforward from the surjectivity of $\phi$ because if $f\circ\phi = 0$ for some $f$ , then by surjectivity of $\phi$ it means that $f$ was zero in the first place. In other words the kernel of $\phi^*$ is reduced to 0.
|
"proof of algebraic independence of elementary symmetric polynomials" is owned by lalberti.
|
|
(view preamble | get metadata)
Cross-references: reduced, kernel, place, conclusion, injective, algebraically independent, theorem, algebras, morphism, transposition, side, roots, field, fix, surjective, closed, algebraic, variables, functions, symmetric, associates, map, substitution, algebraic closure, quotient field, division ring, proof
This is version 1 of proof of algebraic independence of elementary symmetric polynomials, born on 2007-11-26.
Object id is 10063, canonical name is ProofOfAlgebraicIndependenceOfElementarySymmetricPolynomials.
Accessed 747 times total.
Classification:
| AMS MSC: | 05E05 (Combinatorics :: Algebraic combinatorics :: Symmetric functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![\begin{displaymath}\begin{array}{lccr} \phi^*: & R[S_1,\ldots,S_n] & \to & R[T_1,\ldots,T_n]\ & f & \mapsto & f\circ\phi \end{array}\end{displaymath}](http://images.planetmath.org:8080/cache/objects/10063/js/img2.png "\begin{displaymath}\begin{array}{lccr} \phi^*: & R[S_1,\ldots,S_n] & \to & R[T_1,\ldots,T_n]\ & f & \mapsto & f\circ\phi \end{array}\end{displaymath}")