# Tagged Questions

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### 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 ...
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### 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 ...
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### Elementary Symmetric Polynomials, Roots of cubic polynomials

I'm given $a_1, a_2, a_3$ as roots of the equation $x^3 + 7x^2 - 8x + 3$ and need to find the cubic polynomials with roots $a_1^2, a_2^2, a_3^3$ and $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}$. ...
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### Are the elementary symmetric polynomials “unique”?

The elementary symmetric polynomials are interesting in that they generate the set of symmetric polynomials, in the sense that every symmetric polynomial is some polynomial applied to the elementary ...
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### Symmetric function theorem and Galois Theory — How deep is the connection?

By symmetric function theorem in the title, the fundamental theorem of symmetric polynomials is meant: Any symmetric polynomial has a unique representation as a polynomial in the elementary symmetric ...
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### Are the coefficients of $\text{minpoly}(\alpha + \beta)$ polynomials in the coefficients of $\text{minpoly}(\alpha)$ and $\text{minpoly}(\beta)$?

Suppose you are given an algebraic field extension $L \supset K$ and $\alpha,\beta \in L$ with $f(X) = \text{minpoly}_K(\alpha)(X)=a_0+...+a_{m-1} X^{m-1}+X^m$ and ...
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### 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 ...
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### Galois Theory References

This may not initially be a well posed question, but I'm looking for a good reference on Galois theory that covers it from the viewpoint of the symmetry in roots of an irreducible polynomial and not a ...
I'm reading Exploratory Galois Theory by John Swallow. On page 123 he gives the following remark / alternate proof of the fundamental theorem of symmetric polynomials: Let $K$ be a field and $L$ ...
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 ...