# 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^{} $K$ of $Q$.

Consider the substitution map that associates to values ${t}_{1},\mathrm{\dots},{t}_{n}\in {K}^{n}$ the symmetric functions in these variables ${s}_{1},\mathrm{\dots},{s}_{n}$.

$$\begin{array}{cccc}\varphi :\hfill & \hfill {K}^{n}\hfill & \hfill \to \hfill & \hfill {K}^{n}\\ & \hfill ({t}_{i})\hfill & \hfill \mapsto \hfill & \hfill ({s}_{i})\end{array}$$ |

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

$${X}^{n}+\sum _{i}{v}_{i}{X}^{i}={\mathrm{\Pi}}_{i}(X+{t}_{i})$$ |

And by developing the right-hand side we get ${v}_{i}={s}_{i}$.

Then we consider the transposition^{} morphism of algebras ${\varphi}^{*}$ :

$$\begin{array}{cccc}{\varphi}^{*}:\hfill & \hfill R[{S}_{1},\mathrm{\dots},{S}_{n}]\hfill & \hfill \to \hfill & \hfill R[{T}_{1},\mathrm{\dots},{T}_{n}]\\ & \hfill f\hfill & \hfill \mapsto \hfill & \hfill f\circ \varphi \end{array}$$ |

The capital letters are there to emphasize the ${S}_{i}$ and ${T}_{i}$ are variables and $R[{S}_{1},\mathrm{\dots},{S}_{n}]$ and $R[{T}_{1},\mathrm{\dots},{T}_{n}]$ 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, ${\varphi}^{*}({S}_{i})$ is the ${i}^{th}$ symmetric function in the ${T}_{i}$, and ${\varphi}^{*}$ is clearly a morphism of algebras.

The conclusion^{} is then straightforward from the surjectivity of $\varphi $ because if $f\circ \varphi =0$ for some $f$, then by surjectivity of $\varphi $ it means that $f$ was zero in the first place. In other words the kernel of ${\varphi}^{*}$ is reduced to 0.

Title | proof of algebraic independence of elementary symmetric polynomials |
---|---|

Canonical name | ProofOfAlgebraicIndependenceOfElementarySymmetricPolynomials |

Date of creation | 2013-03-22 17:38:28 |

Last modified on | 2013-03-22 17:38:28 |

Owner | lalberti (18937) |

Last modified by | lalberti (18937) |

Numerical id | 4 |

Author | lalberti (18937) |

Entry type | Proof |

Classification | msc 05E05 |