# Tagged Questions

Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.

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### Is there a natural link between symmetric polynomials and symmetric algebra?

Let $R$ be a commutative ring and $R[X_1,...,X_n]^{S_n}$ be the ring of symmetric polynomials. I have learned some basic properties of this ring and the results are really similar to those by ...
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### Find the coefficients in $(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 = s_1^2s_2^2 + as_1^3s_3 + bs_1s_2s_3 + cs_2^3+ds_3^2$

Use evaluation homomorphisms $F[x_1,x_2, \dots, x_n] \to F$ to obtain the coefficients in: $$(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 = s_1^2s_2^2 + as_1^3s_3 + bs_1s_2s_3 + cs_2^3+ds_3^2$$ where the $s_i$ ...
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### A question on a proof concerning resultants I don't quite get

http://planetmath.org/proofthatsylvestersmatrixequalstheresultant Heres a link to a proof I found concerning the relation between Sylvesters matrix and resultants. Most of it makes sense. I do have ...
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### System of equations with 2 parameters

I have no idea how even to start! \begin{align*} (u^2+v^2)(u+v)&=15uv \\ (u^4+v^4)(u^2+v^2)&=85u^2v^2 \end{align*}
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### Coefficients of Lagrange resolvent

I'm trying to make sense of some things I read about Galois theory. Let $p$ be a monic polynomial of degree $n$ with known coefficients $a_i$ and unknown roots $x_i$: \begin{alignat*}{2} p(X) &= (...
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### Solving system when terms have both variables

$$x^3-3y^2x=-1$$ $$3yx^2 -y^3=1$$ This was the real part and imaginary part on a previous question I asked, instead of the system it was easier to just use polar coordinates to solve, but if this was ...
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### Find the equation which has key root $x=\sqrt{a}+\sqrt{b}+\sqrt{c}$

In my last question which was Proving $x=\sqrt{a}+\sqrt{b}$ is the key root to solve $x^4-2(a+b)x^2+(a-b)^2=0$ ,I could find the coefficients(were very easy) of fourth-degree equation, so I went to ...
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### How can I convince students a certain polynomial equation is symmetric?

How can I convince students that $p(x)=0$ is a symmetric equation if they ask me, where $p(x)$ is polynomial of degree $n$ with reals coefficients. For example : $A(x)=2x^4-9x^3+8x^2-9x+2=0$ is ...
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### Finding a generalized form for taking the n$^{th}$ derivative of a falling factorial

I would like to make $$\frac{d^n}{dx^n}[(x)_c] = n! \times e_{c-n}(x,x-1,\cdots,x-c+1)$$ Where $e_{c-n}(x,x-1,x-2,\cdots,x-c+1)$ is the elementary symmetric polynomial function But lets say that ...
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### A question about the elementary symmetric polynomial

I have asked this question and have come up with a possible answer $$\frac{d^j}{dx^j}[\frac{(x)_c}{j!}] = e_{c-j}(x,x-1, \cdots ,x-c+1)$$ My first question is, how can I prove this? It seems trivial ...
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### Question about primes of polynomial type.

It is well known that $50$ % of the primes are of the form $x^2 + y^2$. Many variants exists where a rational amount of primes is of some integer polynomial form. But I wonder ; are there integer ...
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### $a_1^k+a_2^k+\ldots+a_n^k$ integer implies all integers?

Let $n$ be a positive integer, and let $a_1,\ldots,a_n$ be rational numbers. Suppose that $a_1^k+a_2^k+\ldots+a_n^k$ is an integer for all positive integers $k$. Is it true that $a_1,a_2,\ldots,a_n$ ...
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