# 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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### 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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### A Formula of Jet Bundles in Gelfand's Book

In Gelfand's book Discriminants, resultants, and multidimensional determinants he gives a formula: $$J_{1}(L)\cong J_{1}(\mathcal{O}_{X})\otimes L$$ Here $L$ is a line bundle on some complex variety. ...
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### Where does the plus-minus come from in the quadratic formula? [closed]

In the formula $$x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ for solving quadratic equations, where does the $\pm$ come from? The square root already results in both a positive and negative term, ...
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### Discriminant of odd cyclic Galois extension is not a power of 2

Is there a way to prove that, given a Galois cyclic extension $\mathbb{Q} \subset F$ with odd order, there exists a prime $p \neq 2$ such that $p|\Delta_{F}$ ? I'm actually trying to prove that the ...
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