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.
\[R[x_1, \ldots, x_n]^{S_n} = R[s_1, \ldots, s_n]\]