1
vote
2answers
40 views

Constructing expressions

Suppose you are given all the elementary symmetric functions of $n$ variables $x_1,x_2,...,x_n$ and two rational functions $A(x_1,x_2,...,x_n)$ and $B(x_1,x_2,...,x_n)$ in the same $n$ variables that ...
0
votes
0answers
25 views

Unique “splitting fields” for infinite polynomials

When creating the splitting field of a polynomial we can without loss of generality assume it has constant coefficient $1$ and splits as $\prod_{k=1}^n(1-z_kT)$. The splitting field is obtained by ...
1
vote
0answers
37 views

Independent indeterminate roots and coefficients of a polynomial

Dummit and Foote, Section 14.6: "If the roots of a polynomial $f(x)$ are independent indeterminates over a field $F$, then so are the coefficients of $f(x)$." This is meant to complete the converse of ...
0
votes
0answers
61 views

System of equations and Abel theorem

Consider this system of 3 equations to be solved in x,y and z: $a x^m=(y+z)^n$ $by^m=(x+z)^n$ $cz^m=(x+y)^n$ The parameters $(a,b,c)$ and the unknown $(x,y,z)$ are all in $ℝ₊$. Also, m and n are ...
2
votes
1answer
159 views

Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions

A rational function $f$ in $n$ variables is a ratio of $2$ polynomials, $$f(x_1,...x_n) = \frac{p(x_1,...x_n)}{q(x_1,...x_n)}$$ where $q$ is not identically $0$. The function is called symmetric if ...
2
votes
2answers
119 views

Why is this corollary a corollary? (Field extensions and symmetric polynomials.)

In Stewart and Tall's book on Algebraic Number Theory, they give a theorem of Newton: Theorem 1.12. Let $R$ be a ring. Then every symmetric polynomial in $R[t_1, \ldots, t_n]$ is expressible as a ...