Symmetric Functions

Using the fundamental theorem on symmetric functions, prove that an integer rational symmetric function of degree $m$ of the roots of the equation

is rationally expressible in terms of $p_{1}, p_{2}, \ldots , p_{m}$.

Using the result of the preceding problem (see problem 2666), prove that the power sums of the roots $S_{1}, S_{2}, \ldots , S_{m}$ of the equations

and

are the same. Here $m \leqslant n$.

Drawing on the result of the two preceding problems (see problems 2666, 2667), prove that, for $m \leqslant n$, the power sums of the roots of the equation

satisfy the formula

By multiplying the left-hand side of the equation $x^{n}+p_{1}x^{n-1}+p_{2}x^{n-2}+\ldots +p_{n}=0$ by a suitable power of $x$, obtain, for $m > n$, the equality for the power sums of the roots of the equation:

Find the value of the indicated symmetric function of the roots of the given equation: $x^{5}-3x^{3}-5x+1=0$; $\sum x_{\nu }^{4}$.

Find the value of the indicated symmetric function of the roots of the given equation: $x^{3}+3x^{2}-x-7=0$; $\sum x_{\nu }^{6}$.

Find the value of the indicated symmetric function of the roots of the given equation: $x^{4}+ax^{3}+bx^{2}+cx+d=0$; $\sum x_{\nu }^{3}x_{\mu }$; $\nu \neq \mu$.

Find the value of the indicated symmetric function of the roots of the given equation: $x^{3}-x-1=0$; $\sum x_{1}(x_{2}-x_{3})^{2}$.

Find the value of the indicated symmetric function of the roots of the given equation: $x^{4}-5x^{2}-2x+1=0$; $\sum (x_{1}-x_{2})^{2}(x_{3}-x_{4})^{2}$.

Find the value of the indicated symmetric function of the roots of the given equation: $x^{n}+x^{n-1}+\ldots +x+1=0$; $\sum x_{1}^{3}x_{2}^{3}x_{3}^{3}$; $n > 8$.

Find the value of the indicated symmetric function of the roots of the given equation: $x^{5}-4x^{3}+x^{2}+3x+1=0$; $\sum x_{1}^{2}x_{2}^{2}x_{3}x_{4}$.

$x^{n}+(a+b)x^{n-1}+(a^{2}+ab+b^{2})x^{n-2}+\ldots +(a^{n}+a^{n-1}b+\ldots +b^{n})=0$; find $S_{m}$ for $m \leqslant n$.

$x^{n}+ax^{n-1}+\dfrac {a(a+1)}{1 \cdot 2}x^{n-2}+\ldots +\dfrac {a(a+1)\ldots (a+n-1)}{1 \cdot 2 \ldots n}=0$. Find $S_{m}$ for $m \leqslant n+1$.

Prove Waring's formulas, giving expressions for $S_{m}$, the sum of the $m$-th powers of the roots of the equation $x^{n}+p_{1}x^{n-1}+\ldots +p_{n}=0$, in terms of the coefficients of the equation. These formulas can be written in the form

In each of the sums on the right-hand side, the indices $\mu , \nu , \lambda , \ldots$ are positive, and their sum equals $m$.

For the cubic equation $x^{3}+ax^{2}+bx+c=0$, evaluate the following symmetric functions: $\dfrac {x_{1}^{2}+x_{2}^{2}}{x_{1}+x_{2}}+\dfrac {x_{1}^{2}+x_{3}^{2}}{x_{1}+x_{3}}+\dfrac {x_{2}^{2}+x_{3}^{2}}{x_{2}+x_{3}}$; $\dfrac {x_{1}^{2}+x_{2}x_{3}}{x_{2}+x_{3}}+\dfrac {x_{2}^{2}+x_{1}x_{3}}{x_{1}+x_{3}}+\dfrac {x_{3}^{2}+x_{1}x_{2}}{x_{1}+x_{2}}$.

For the cubic equation $x^{3}+ax^{2}+bx+c=0$, find the area of the triangle with sides $x_{1}, x_{2}, x_{3}$.

Represent, as a polynomial in $x$, the following fractional function of a root of the given equation (rationalizing the denominator): $\dfrac {6+5x}{1+x+x^{2}}$; $x^{3}-2=0$.

Represent, as a polynomial in $x$, the following fractional function of a root of the given equation (rationalizing the denominator): $\dfrac {1}{x^{3}-x+2}$; $x^{4}-5=0$.

Represent, as a polynomial in $x$, the following fractional function of a root of the given equation (rationalizing the denominator): $\dfrac {1}{1+2x+x^{2}}$; $x^{3}-x-1=0$.

Represent, as a polynomial in $x$, the following fractional function of a root of the given equation (rationalizing the denominator): $\dfrac {1}{a+bx+cx^{2}}$; $x^{3}-N=0$.

Determine whether the polynomials $\varphi (x)=x^{3}-3x^{2}+x+1$, $\psi (x)=2x^{3}-4x+2$ have a common root.

Determine whether the polynomials $\varphi (x)=x^{2}+x+1$, $\psi (x)=x^{3}+x-1$ have a common root.

Determine whether the polynomials $\varphi (x)=x^{5}+x^{2}+1$, $\psi (x)=x^{5}-x^{3}+2$ have a common root.

For what $\lambda$ and $\mu$ do the equations $x^{3}-6x^{2}+\lambda x-3=0$, $x^{3}-x^{2}+\mu x+2=0$ have two common roots?

Solve the system of equations:

Solve the system of equations:

Eliminate $x$ from the system of equations:

Eliminate $x$ from the system of equations:

Eliminate $x$ from the system of equations:

Find the discriminant of the equation $x^{n}+a=0$.

The same for the equation $x^{3}+px^{2}+qx+r=0$ (see problem 2695).

The expression for the discriminant $D(a_{0}, a_{1}, \ldots , a_{n})$ contains the term $Ma_{0}^{n-1}a_{n}^{n-1}$. Find $M$.

Prove the theorem: if all the roots of an equation with integer coefficients $x^{n}+p_{1}x^{n-1}+\ldots +p_{n}=0$ have absolute value equal to unity, then they are roots of unity. (Kronecker.)