Galois Theory HT25, Symmetric functions
Flashcards
@Define a symmetric polynomial in $n$ variables with coefficients in $R$.
An element of $R[x _ 1, \ldots, x _ n]^{S _ n}$.
@Define the $k$-th symmetric polynomial in $n$ variables, denoted $s _ k$.
\[s _ k := \sum _ {i _ 1 < i _ 2 < \cdots < i _ k} \prod^k _ {j = 1} x _ {i _ j} \in \mathbb Z[x _ 1, \ldots, x _ n]\]
@State the fundamental theorem of the theory of symmetric functions.
\[R[x _ 1, \ldots, x _ n]^{S _ n} = R[s _ 1, \ldots, s _ n]\]