# fundamental theorem of symmetric polynomials

Every symmetric polynomial^{} $P({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ in the indeterminates ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}$ can be expressed as a polynomial^{} $Q({p}_{1},{p}_{2},\mathrm{\dots},{p}_{n})$ in the elementary symmetric polynomials
${p}_{1},{p}_{2},\mathrm{\dots},{p}_{n}$ of ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}$. The polynomial $Q$ is unique, its coefficients are elements of the ring determined by the coefficients of $P$ and its degree with respect to ${p}_{1},{p}_{2},\mathrm{\dots},{p}_{n}$ is same as the degree of $P$ with respect to ${x}_{1}$.

Title | fundamental theorem of symmetric polynomials |
---|---|

Canonical name | FundamentalTheoremOfSymmetricPolynomials |

Date of creation | 2013-03-22 19:07:40 |

Last modified on | 2013-03-22 19:07:40 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 13B25 |

Classification | msc 12F10 |

Synonym | fundamental theorem of symmetric functions |