Alternierende Gruppe/Polynomring/Invariantenring/Fakt

Es sei ${}K$ ein Körper der Charakteristik ${}\operatorname {char} (K)\neq 2$ . Die alternierende Gruppe ${}A_{n}$ operiere natürlich auf ${}V=K^{n}$ .

Dann ist

${}K[V]^{A_{n}}=K[V]^{S_{n}}\oplus K[V]_{\operatorname {sgn} }^{S_{n}}=K[E_{1},\ldots ,E_{n}]\oplus K[E_{1},\ldots ,E_{n}]\cdot \triangle \,.$

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