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Discriminant - Discriminant of an algebraic number field
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Discriminant, Discriminant - Discriminant of a conic section, Discriminant - Discriminant of a polynomial, Discriminant - Discriminant of a quadratic form, Discriminant - Discriminant of an algebraic number field

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Discriminant - Discriminant of an algebraic number field: Encyclopedia II - Discriminant - Discriminant of a polynomial

The discriminant of a polynomial is a number which can be easily computed from the coefficients of the polynomial and which is zero if and only if the polynomial has a multiple root. For instance, the discriminant of the polynomial ax2 + bx + c is b2 − 4ac. For the general definition, suppose p(x) = xn + an−1xn−1 + ... ...

There is a substantive generalisation, to quadratic forms Q over any field K of characteristic ≠ 2. These can be written as a sum of terms aiLi2 where the Li are linear forms and 1 ≤ i ≤ n where n is the number of variables. Then the discriminant is the product of the ai, taken in K/K2, and is then well-defined (i.e., up to squares). A more invariant way to say this is as ( ...

For a conic section defined by the real polynomial: ax2 + bxy + cy2 + cx + ey + f= 0, the discriminant is equal to b2 − 4ac, and determines the shape of the conic section. If the discriminant is less than 0, the equation is of an ellipse or a circle. If the discriminant equals 0, the equation is that of a parabola. If the discriminant is greater than 0, the equation is that of a hyperbola. This formula will ...

Discriminant - Discriminant of an algebraic number field

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