Proposition 6

— Let $f \in A[X]$ be a monic polynomial of degree n, and let P be a symmetric polynomial in $X_1, ..., X_n$ with coefficients in A. Then there exists precisely one element a of A with the following property: (FS) For any ring homomorphism $\rho : A \to B$ and decomposition $^0f(X) = \prod_{i=1}^n (X - \xi_i)$ in B[X] we have $\rho(a) = P(\xi_1, ..., \xi_n)$.

Write $f = X^n + \sum_{k=1}^n a_k X^{n-k}$, then by Th. 1, IV, p. 62 there exists a polynomial $\Pi$ in n indeterminates with coefficients in A, such that $P = \Pi(s_1, ..., s_n)$. Put $a = \Pi(-a_1, a_2, ..., (-1)^n a_n)$. Under the hypothesis (FS) we have

whence

This proves the existence of an element a satisfying (FS). The uniqueness of a follows from the Cor. to Prop. 5, because we have $a \cdot 1 = P(x_{1,f}, ..., x_{n,f})$ in the universal decomposition algebra $E_f$.

With the notation of Prop. 6 one sometimes writes $a = P^*(f)$. Here are some examples.