I've become confused with a certain sentence of an abstract algebra text.
The author is talking about the Field extension $ℚ(x_1, x_2, ... , x_n)/ℚ$, where each $x_i$ is one of the roots of a polynomial of degree $n$.
He follows to explain that "the most important property of $ℚ(x_1, x_2, ... , x_n)$ is that it is symmetric with respect to $x_1, x_2, ... , x_n$, in the sense that any permutation $*$ of $x_1, x_2, ... , x_n$ extends to a bijection * of $ℚ(x_1, x_2, ... , x_n)$ defined by
$$* f(x_1, ... , x_n) = f(*x_1, ... , *x_n)$$
for each rational function $f$ of $x_1, ... , x_n$. Moreover, this bijection * obviously satisfies
$$*(f + g) = *f + *g,$$ $$*(fg) = (*f)(*g)$$
The issue is that I'm not really understanding what this permutation $*$ is, nor why it has the properties that the author claims it has.
I also don't get what he means by "symmetric". He was talking before about how each coefficient $a_i$ of the polynomial is a function of the roots. He called them "elementary symmetric functions". I guess -and might be completely wrong here- that the name has something to do with the fact that substituting a certain root $x_j$ with $x_k$ in any of these 'symmetric functions' wouldn't mess up the equation.
I don't know much about Group theory, so forgive me if I ask too many questions about the notation/terminology of the replies
I would really appreciate any help/thoughts!