I was looking the proof of fundamental theorem on symmetric polynomials from Lang's Algebra (see this, p.190-192). I didn't understood one thing from proof and one question came to mind from statement of theorem.
Question 1. Why weight of the monomial $X_1^{v_1}\cdots X_n^{v_n}$ is defined as $v_1+2v_2+\cdots + nv_n$, instead of just $v_1+v_2+\cdots + v_n$? I think, even-though we define this way - $v_1+v_2+\cdots + v_n$ - then still induction argument may work in the proof. .
Question 2. In statement of Theorem 6.1, it is said that $g$ is of degree at most degree of $f$. My question is that, isn't it necessary that $g$ should of degree equal to $d$? What is example in this case?
Notice that the conclusion of the theorem is: $f(t) = g(s_1, s_2, \ldots, s_n)$, where we have substituted $s_i$ for $X_i$. The polynomial $s_i$ is the elementary symmetric polynomial of degree $d$, so $X_i$ should have a weight of $d$, so that the weight of $g$ before substitution will match the degree of $g$ after substitution. The purpose of the weights is to keep track of what the resulting degree of $g$ will be, before you actually make the substitution.