# 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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### Solving simple quadratic - Wolfram Alpha confusion?

I have the following quadratic $$(2\sqrt 2 - 2)x^2 + \sqrt8 x + (1+\sqrt 2)=0$$ Now the discriminant of this is $0$, so it has one real repeated root. A plot on Desmos confirms this. However, ...
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### Finding “determinant” for rank-4 tensor that's not for linear algebra

Please pardon me if I'm not using the right terminology. I'm using a rank-4 tensor T ($n \times n \times n \times n$) to store connections among n nodes through tetrodes (things connecting to 4 nodes ...
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### Discriminant when graph lies above or below the x axis.

Suppose a quadratic equation has been given where the a value (ax^2 + bx + c) is a positive and it has been said that the graph of the equation lies above the x-axis- what is the discriminant? For ...
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### Factorisation of fundamental discriminants

$D$ is a fundamental discriminant if $D\equiv 1\pmod 4$ and $D$ is square-free or $D=4m,$ where $m\equiv 2,3 \pmod 4$ and $m$ is square-free. On wikipedia I found the following characterisation: ...
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### What happens to the group structure of an elliptic curve over a field when the discriminant = 0?

Working on a question for a number theory class. So, basically, it asks us what happens to the group structure of an elliptic curve over a field if the discriminant is equal to zero? So, basically,...
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### Discriminant Integral closure of ring

I have trouble to understand how to generalize the definition of the discriminant $Disc(B/A)$ where $A$ is the ring of integer in some number field $K$ and $B$ the integral closure in some finite ...
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### Determining the nature of quadratic equation

If $a,b,c$ are real numbers and $a+b+c =0$ then how to prove that the equation $$4ax^2+3bx+2c$$ has two real roots. I just know that for real roots the quadratic equation should have its Discriminant ...
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### Primes representable by equivalent binary forms

So the question I have is this , if a prime p is representable by two primitive positive definitive binary quadratic forms f(x,y) and g(x,y) of the same discriminant, are they necessarily equivalent? ...
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### Finding the value of k that satisfies the discrimant

I am struggling to do this question Find the valus of $k$ so that $kx^2-2+kx+x=0$ has discriminant $Δ= -24$ I tried to use the formula $b^2-4ac$ but my answer is completely wrong. Any ideas ...
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### Multivariable Resultant

I want to compute the resultant of $f_1,\cdots, f_m \in K[x_1,\cdots x_n]$ where $f_1, \cdots,f_m$ are forms of degrees $d_1,\cdots ,d_n$ respectively. Does anyone have any idea on how could I do this?...
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### Does this fact about the minimal polynomial give an efficient diagonalizability criterion?

In a very nice paper "When Is a Linear Operator Diagonalizable?" by Marco Abate (Amer. Math. Monthly 104 (1997), 824-830) I found the following nice description of the minimal polynomial $\mu(T)$ of ...
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### Properties of a polynomial with zero discriminant

In Wikipedia it says that "The discriminant of a polynomial over an integral domain is zero if and only if the polynomial and its derivative have a non-constant common divisor." How does one ...
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### Confusion about the definition of ideal (of ring of algebraic integers)

I am studying algebraic number theory and am confused about the following lemma. We prove that if $I \subset O_K$ a non-zero ideal then $$\textrm{disc}(I) = \textrm{disc}(O_K)\cdot N(I)$$ Then ...
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### Matrix Notation Form of Roots of a Quadratic Equation

We know that the quadratic equation $$f(x)=ax^2+bx+c=0$$ has roots $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=-\frac b{2a}\pm \frac 1a\sqrt{-\left(ac-\frac {b^2}4\right)}$$ Also, $f(x)$ can be written in ...
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Let $f(X) = X^2 + bX + c \in K[X]$ be a quadratic polynomial whose discriminant is $\delta^2 = b^2 - 4c$. I understand that if $\alpha_1$ and $\alpha_2$ are the roots of $f$, then $\delta = \alpha_1 - ... 2answers 108 views ### Proving$a^4+b^4+c^4+(\sqrt {3}-1)(a^2 b c+a b^2 c+a b c^2 )\ge \sqrt {3} (a^3 b+b^3 c+c^3 a)$for real$a$,$b$,$c$If$a$,$b$,$c$are real numbers, I have to prove: $$a^4+b^4+c^4+(\sqrt {3}-1)(a^2 b c+a b^2 c+a b c^2 )\ge \sqrt {3} (a^3 b+b^3 c+c^3 a)$$ Since $$a^4+b^4+c^4 \ge abc(a+b+c)$$ then it is enough ... 1answer 43 views ### Let$f \in F[x]$, then discriminant of$f$lies in$F$. Let$\alpha_1, \ldots, \alpha_n$be the roots of$f$in some splitting field$K$. Then define$disc_K(f) = \prod_{i<j} (\alpha_i - \alpha_j)^2$. Is it true$disc_K(f) \in F$? I used to prove$...
I am currently working with positive-definite, reduced, primitive, integral binary quadratic forms, and I have noticed something interesting. Conjecture: Let $Q$ be a form of non-fundamental ...