# Questions tagged [discriminant]

Discriminant of a polynomial $\;P\left(x\right) = a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{n}x^{n} \neq 0\,$ is defined as \begin{align} \Delta &= a_{n}^{2n-2}\prod_{ i < j } \big( r_i - r_j \big)^{2} = \left(-1\right)^{n\left(n-1\right)/2} a_{n}^{2n-2}\prod_{ i \neq j } \big( r_i - r_j \big) \end{align} where $\,r_1,\dots,r_n\,$ are roots of $P\left(x\right)$ (counting multiplicity)

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### Inequality with x,y,z fractions $\frac{x}{y}+\frac{y}{z+x}+\frac{z}{x}\ge 2$

If $x,y,z>0$, show: $$\frac{x}{y}+\frac{y}{z+x}+\frac{z}{x}\ge 2$$ I expand and to prove $$x^3 - 2 x^2 y + x^2 z + x y^2 - x y z + y z^2\ge 0$$ I don't know how to do this.
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### What is the condition that a cubic equation $x^3+ax^2+bx+c=0$ has exactly three positive real root?

What is the condition that a cubic equation $x^3+ax^2+bx+c=0$ has exactly three positive real root? If $G^2+4H^3<0$ then it has three real roots (where $G=c-ab/3+2a^2/27$, $H=b-a^2/9$). Then what ...
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### Connection between an algebraic invariant and a maximisation issue

I came to be interested by the following rational function : $$f(x)=\dfrac{(x^2-x+1)^3}{x^2(x-1)^2}\tag{1}$$ while writing this answer ; I discovered that $f$ is connected to rather deep features of ...
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### Discriminant of $X=Z(f)$ in terms of join, intersection and projection.

Say we have $X\subset \mathbb{CP}^{n}$ a projective variety, we may even assume it is a hypersurface for convenience. Let $\pi:\mathbb{CP}^{n}\to\mathbb{CP}^{n-1}$ be the standard projection. I am ...
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### Roots of quadratic eqaution lies in an AP

I tried finding the common difference between the roots but didnt know what to do next. if someone could tell me how to solve this or give a headstart itll be a lot helpful
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### Finding discriminant of a monic polynomial.

I have now engaged in studying Galois Theory from NPTEL online lecture series which encompasses Finite Fields and Galois Theory. While watching the $48$-th lecture on Discriminant of a Polynomial a ...
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### Writing Cubic Equation in terms of discriminant (with possible shifts and translations)

So I noticed this fact for the following fact for quadratic equations. I need one notation that if one equation can be gotten from another through a shift or scaling of variable then I will denote ...
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### Find the ring of integers of $\mathbb{Q}(\theta)$

I was trying to find the ring of integers of $\mathbb{Q}(\theta)$, where $\theta^3 -2\theta + 2 = 0$. I compute the discriminant of the basis $\{1, \theta, \theta^2\}$, but unfortunately it is $-4*19$,...
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### Can we find the discriminant of an Equation of any degree?

Some days ago I was solving a question which gave me a hard time. After doing some research I found out that it required the Discriminant of a Cubic Equation. I looked up to the internet and I found ...
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### $\mathbb{Q}(\sqrt{n})$ is Contained in the Cyclotomic Field of the $4n$'th Primitive Root of Unity.

From Silverman and Tate, Rational Points on Elliptic Curves. Exercise 6.1 Let $\zeta'$ be the $4n$'th primitive root of unity. Use (c) to prove that $\mathbb{Q}(\sqrt{n})$ is contained in the ...