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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how would i find the domain for this prove question using the discriminant

How would I prove that 3x^2-x+7=0 goes for all real x I've already tried discriminant method b^2-4ac but it's still incorrect and using the quadratic formula gives me an error.
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Discriminant of depressed cubic [duplicate]

If the cubic equation $x^3+px+q$ has roots $\alpha , \beta , \gamma $ then we know that $\alpha + \beta + \gamma =0 $, $\alpha \beta + \alpha \gamma + \beta \gamma =p $ and $\alpha \beta \gamma =-q $. ...
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Can the jacobi symbol be used for the statement "n is represented by some quadratic form of discriminant d iff 4n is a square mod d"

We've been using the above statement repeatedly in a number theory course, but to find all primes that are represented by a quadratic binary form of discriminant d, we've been using $$(\frac{d}{4p}) = ...
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Discriminant of $x^{n+1}+x$

I want to compute the discriminant of $x^{n+1}+x$. It's easy to see the roots of this polynomial are $0$ and $e^{(2k+1)\pi i/n},k=0,1,\dots,n-1$. But it's still quite difficult, if we compute all $\...
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squarefree discriminant

Let $F$ be finite field of characteristic not equal to 2. For a generic polynomial $f$ in the function field $\mathbb{F}(u)[t]$, I'm able to show that $f$ is separable over $\mathbb{F}(u)$ and the ...
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Converse of the invariance theorem of conics

Let $Q(x,y)=ax^2+2hxy+by^2+2gx+2fy+c$ be a conic in $2$-dimensional Euclidean space $\mathbb{E}^2$ and also let $$\tau= \text{tr}\left(\begin{array}{cc} a &h\\h&b \end{array}\right),$$ $$\...
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What is $d(\mathbb{Q}(\sqrt{2},\sqrt{3}))$?

$K=\mathbb{Q}(\sqrt{2},\sqrt{3})=\mathbb{Q}(\sqrt{2}+\sqrt{3})$ and $[\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}]=[\mathbb{Q}(\sqrt{2}+\sqrt{3}):\mathbb{Q}]=4$. We also know the conjugates of $\sqrt{2}+\...
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What are the necessary and sufficient conditions for the cubic equation to have at least 1 positive real root?

What are the necessary and sufficient conditions for the cubic equation to have at least 1 positive real root? I'm just dealing with the $2$ simplest cases. Case-1: $$x^3+px+q=0$$ where, $p<0$ and ...
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Discriminant of a polynomial: a question about definition

Let $f(x)$ be a non-constant polynomial of degree $n$ over a field $F$ and $r_1,r_2,\ldots, r_n$ be its roots in some extension of $F$. The discriminant of $f(x)$ is a symmetric expression in its ...
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Discriminant in terms of roots

Discriminant of a polynomial has different expressions. Suppose we want to define (and confine) with the definition via roots of a polynomial. Then I have the following question. ($1$) If $f(x)=x^n+ ...
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What is the number of solution of equation of locus when a straight line is tangent to the locus.

Find the values of t if a straight line y=t is the tangent to the locus of point P , $$x^2+y^2+4x-6y-3=0$$ where t is a constant. This is an exam question from a school. The solution in the answer ...
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Discriminants of quartic fields are $k$-power free for some $k$?

Let $K/\mathbb{Q}$ be a quadratic number field. Then the discriminant of $K$ is either $d$ or $4d$ for some squarefree integer $d$, so the discriminant is never divisible by an odd square. I am ...
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Hungerfords definition of the discriminant

Hungerfords Algebra Definition 4.4: Let $K$ be a field with char$K \neq 2$ and $f \in K[x]$ be a polynomial of degree $n$ with $n$ distinct roots $u_1,..u_n$ in some splitting field $F$ of $f$ over $K$...
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A closed form relation to compute the spectral norm of a 2x2 real matrix

I am reading this book (page 215) and I found in the bottom of this page an interesting relation that I want to use. The relation is $$||A||_2 = \sqrt{\frac{||A||_F^2 + \sqrt{||A||_F^4 - 4 (\mathrm{...
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Discriminant of $\mathbb{Q}$

We can calculate the discriminant of an algebraic number field $\mathbb{Q}(\sqrt{d})$, with $d$ not a square by: $$ disc(\mathbb{Q}(\sqrt{d}))=\det(\sigma_i(\alpha_j))^2 $$ (ref: p. 18 "Number ...
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Discriminant of a number field in terms of variable

I want to determine the discriminant of the number field generated by the polynomial $$x^8-24tx^4+36t^2$$ in terms of the variable $t$, a square-free integer. Is this possible?
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Why are the discriminants for a given polynomial & its resolvent polynomial congruent?

I have been looking into the formulae for the cubic & quartic polynomials, using the resolvent method to come up a corresponding resolvent polynomial of a lesser degree. It seems that it for the ...
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Why is the class number $h(-16)=1$?

For an integer $d<0$, we defined the class number of $d$, denoted $h(d)$ as the number of reduced positive definite binary quadratic forms with discriminant $d.$ We quoted a theorem (Baker, Stark ...
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Discriminant of Elliptic curve over binary field

Let $E$ be an elliptic curve over field of $char=2$. The nonsupersingular curve is given by Weierstrass equation $E:y^2+xy=x^3+ax^2+b$ where the discriminant is $b$. On the other hand, the ...
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How to compute the formula of discriminant of generic Weierstrass equation of elliptic curve

Let $E$ be an elliptic curve defined over a field $k$ given by Weierstrass equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6\ .$$ Every standard book of elliptic curve includes the formula for ...
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Definition of discriminant of local field

Let $K$ be a local field, and $L$ be an finite extension of $K$. Then, $L$ is also local field. Then, what is the definition of discriminant of extension L/K ? Discriminant of extension of $\mathbb Q$ ...
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Generalisation of discriminant to roots of given multiplicity

Let $p$ be a polynomial over the complex numbers. The discriminant of $p$ is zero if and only if it has a multiple root. I was wondering: For $n\geq 3$, does there exist a function $f_n$ such that $...
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Why 16 in the discriminant of short weierstrass elliptic curve [duplicate]

Let $E:y^2=x^3+ax+b$ be a elliptic curve over finite field $\mathbb{F}_p$ with $char\neq2,3$. Some books say that discriminant is $-16(4a^3+27b^2)$ and some say $-(4a^3+27b^2)$. My calculation using ...
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On deriving the $T_1=0$ condition for tangents

On this page, the first theorem and proof detail how the standard method to find the equation of the tangent to a conic ($S_1=0$ for a point $(x_1, y_1)$) works. Below equation (12), there's this ...
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the number of solutions to polynomial

The problem is that, for the function $f$: \begin{equation*} f(x)=c_{0} + c_{1}x + c_{2}x^{2} + \ldots + c_{n}x^{n}, c_i \in \mathbb Z \end{equation*} then for $f(x)=0$, how many distinct solutions ...
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Range of values of k such that graph has no stationary point

I am given this graph and I am asked to find the range of values of k such that the graph has no stationary points. $$y=\frac{(x+2)}{x(x+k)}$$ I understand to differentiate the equation and let it “...
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On the intuitive explanation for the discriminant property of conics

In this old post detailing the intuition for the classification of conics based on the sign of the discriminant, the author has considered the coordinates $x$ and $y$ of the point on a conic to become ...
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Is there a situation such that the 'magnitude' of discriminant is important?

For example, we know x^2-x+1 = 0 has discriminant -3, its sign is minus, so it has two different non-real roots. The sign of ...
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Find the maximum of the value $F=x^3y+y^3z+z^3x$

let $x,y,z$ be real number.if $x+y+z=3$,show that $$x^3y+y^3z+z^3x\le \dfrac{9(63+5\sqrt{105})}{32}$$ and the inequality $=$,then $x=?,y=?,z=?$ I can solve if add $x,y,z\ge 0$,also see: Calculate ...
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Unreal root of quadratic equation

Set a positive real number such that $$a^3=6(a+1)$$ Prove that the equation $$x^2 + ax + a^2 -6 = 0$$ there is no real solution Solution attempt: condition : $$a^2 - 4a^2 + 24 <0$$ $$a^2>8$$ $$a&...
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What is the discriminant of $\Phi_{2}(X)$ over $\Bbb Q$?

What is the discriminant of $\Phi_{2^n}(X)$ over $\Bbb Q$ for $n=1$ since $\Phi_2(X)=X+1$ has only one root $-1$? I have calculated $disc(\Phi_{2^n}(X))$ for $n\geq 2$ which matches exactly with what ...
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Discriminant Problems

We all know that the discriminant is the part $b^2-4ac$ of the equation $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ that we use to find the roots of a quadratic equation eg: $ax^2+bx+c=0$ or the part in a ...
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$\displaystyle |g(x)| \leq |f(x)|$ for polynomials

I have got a polynomial: $$f(x) = x^4 - 5x^2 + 5$$ And the condition for a polynomial $g(x)$: $$\forall x \in \mathbb{R}, |g(x)|\leq |f(x)|$$ Prove that $f(x) = a \cdot g(x)$ It's quite easy to see ...
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$\mathbb{Q}(\alpha)$ extension of degree 3 is galois over $\mathbb{Q}$ if and only if discriminant of minimal polynomial of $\alpha$ is square.

I supposed that $\alpha$, $\beta$ and $\gamma$ are the roots of the minimal polynomial in its splitting field. So the discriminant is $(\alpha-\beta)^2(\alpha-\gamma)^2(\beta-\gamma)^2$. If every ...
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An inequality with $x,y,z \ge -1$ and $x+y+z=1$

For $x,y,z \ge -1$ and $x+y+z=1$, Prove: $$ \frac{8x}{yz}+\frac{8y}{zx}+\frac{8z}{xy}+\frac{1}{x^2y^2z^2}+9 \ge \frac{30}{xyz} $$ The difficulties of this qustion are that the constraint condition is $...
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Clairaut differential equations and elliptic discriminants

I was solving this math.SE question, which was asking to solve the Clairaut differential equation $y= xy' - (y')^3$. Just to have nicer signs, I then looked at the equivalent equation $$ y= xy' + (y')^...
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Finding the discriminant of a quadratic equation from the given information on the roots of a quadratic equation

I recently came accross an old question that I solved during my school days. Which is If $\alpha, \beta$ are two real roots of a quadratic equation $ ax^2+bx+c=0 $ and $\alpha+\beta, \alpha^2+\beta^2,...
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Discriminant of q-analogs of positive integers

Computations suggest that the discriminant of the polynomial $[n]_{q^m}=1+q^{m}+\dots+q^{(n-1)m}$ with respect to $q$ is $ \pm {\left( {m{{(mn)}^{n - 2}}} \right)^m}.$ Could you please give me a ...
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Suppose $a^2-4b$ $\neq 0$. Let $\alpha, \beta$ be the (distinct) roots of the polynomial $x^2+ax+b$. Then there is a real number $c$ s.t.

Claim: Let $a,b$$\in$$\mathbb{R}$ and suppose $a^2-4b$$\neq$$0$. Let $\alpha$ and $\beta$ be the (distinct) roots of the polynomial $x^2+ax+b$. Then there is a real number c such that either $\alpha$ $...
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Geometric interpretation of discriminant of a number field (and reference suggestion)

I recently finished a good chunk of Vakil's Algebraic Geometry book (edit: at the time of writing this question I had finished up to chapter 19 and the answer is the content of chapter 21) and am now ...
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How to prove that $\Pi_{1 \leq j \leq p^n, p \nmid j}((\zeta_{p^n}^j)^{p^{n-1}}-1) = p^{p^{n-1}}$?

How do I prove that $\Pi_{1 \leq j \leq p^n, p \nmid j}((\zeta_{p^n}^j)^{p^{n-1}}-1) = p^{p^{n-1}}$? I calculated some examples in Wolfram Alpha where the results were of this specific form, and it ...
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Knowing A and the discriminant, how to find B and C? [closed]

On a Quadratic equation of the form $ax^2+bx+c$, I am given the discriminant ($\Delta=b^2-4ac$) and A. To find B and C, I tried to solve it through $x1$ and $x2$...: $x1 = \frac{-b-\sqrt{\Delta}}{a}$ ...
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How are these two discriminants of polynomials related?

I was wondering the following problem, suppose we have $a,b$ such that $\mathbb Q(a)=\mathbb Q(b).$ Then consider the minimal polynomial of $a$, say $m_a(x)$ and $m_b(x)$ defined analogously. Is it ...
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Intuitive meaning of discriminant as determinant of trace values

Let $A$ be a commutative ring, $n\in\Bbb N$, $R$ be an $A$-algebra which is free $A$-module of rank $n$. If $x_i\in R$ for $i\lt n$, then the discriminant is defined as: $$\operatorname{dis}_{R/A}(x_i:...
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For an inner product space $V_K$, for which fields $K$ is the discriminant is a complete invariant of the form?

The discriminant is an invariant of a general symmetric bilinear form on a vector space $V$ over a field $K$. I'm trying to understand for which fields the discriminant is a complete invariant. First ...
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The discriminant as an invariant for the matrix representation of an inner product related to discriminant of quadratic formula i.e. $b^2-4ac$?

Let $V$ be a vector space equipped with the inner product $<-,->$. Let $A$ and $B$ be marix expressions for $<-,->$ in two different basis for $V$ that are equivalent, so that there exists ...
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Discriminant of a Quadratic Polynomial

I am a little confused about how the author came to this discriminant of this quadratic polynomial. I understand that the discriminant comes from solving for a variable, and the discriminant is ...
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Show this property of primitive quadratic forms

Let $f(x,y),g(x,y)$ be two primitive quadratic forms of discriminant $D < 0$. Then the following statements are equivalent: (i) $f(x,y)$ and $g(x,y)$ are in the same genus i.e. they represent the ...
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Discriminant of $\mathbb{Q}(\sqrt{-13})$

Here's my attempt at calculating the discriminant of $\mathbb{Q}(\sqrt{-13})$. The $\mathbb{Z}$-basis of its ring of integers $R=\mathbb{Z}(\sqrt{-13})$ is $\{e_1,e_2\}:=\{1,\sqrt{-13}\}$ Its complex ...
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Determine the sign of $169\cdot7-49\sqrt{123}$

Determine the sign of the discriminant of the equation $$49x^2-26\sqrt7x+\sqrt{123}=0.$$ The coefficients $a,b$ and $c$ of the equation are: $$a=49,\\b=-26\sqrt7\Rightarrow k=-13\sqrt7,\\c=\sqrt{123}.$...
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