# Question in understanding terminology in Galois Theory from Hungerford Algebra

While self studying algebra from Thomas Hungerford I have following question in understanding terminology on page 253 of textbook.

Consider this line of textbook

Let n be a positive integer, K an arbitrary field, E the subfield of symmetric rational functions in $$K(x_{1},..., x_{n} )$$ and $$f_1, f_2,..., f_n \in E$$ the elementary symmetric functions in $$x_1 , x_2,..., x_n$$ over K.

What does author mean by $$f_1, f_2,..., f_n \in E$$?

Does it mean that range of $$f_{i}$$'s belongs to E?

Clearly, it is a function in variables $$x_{1} ,..., x_{n}$$.

It means that the $$f_i$$'s them-self belong to $$E$$, not their range. The set $$E$$ contains all symmetric rational functions and hence the $$n$$ elementary symmetric functions $$f_1, f_2, \dots, f_n$$ are elements of $$E$$.
(Note that the elementary symmetric functions are called $$e_i$$ instead of $$f_i$$ in the Wikipedia article I linked.)
• It means that the coefficients in the symmetric polynomials are elements of $K$. Sep 24, 2020 at 10:52
Recall that \begin{align*} f_1(x_1,\dots,x_n)&=\sum_{i}x_i,\\ f_2(x_1,\dots,x_n)&=\sum_{i You can check that each of this function $$f_i$$ is symmetric, that is $$f_i(x_{\sigma(1)},\dots,x_{\sigma(n)})=f_i(x_1,\dots,x_n)$$ for all $$\sigma \in S_n$$ and all $$i=1,\dots,n$$. So $$f_1,\dots,f_n\in E$$.