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The discriminant of a quadratic form is invariant under linear changes of variables (that is a change of basis of the vector space on which the quadratic form is defined) in the following sense: a linear change of variables is defined by a nonsingular matrix S, changes the matrix A into , and thus multiplies the discriminant by the square of ...
The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in K / (K ×) 2 (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only "positive, zero, or negative".
In mathematics, a quadratic equation (from Latin quadratus ' square ') is an equation that can be rearranged in standard form as [1] + + =, where the variable x represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.)
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables (,) = + +,where a, b, c are the coefficients.When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form.
Consider a quartic equation expressed in the form + + + + =: There exists a general formula for finding the roots to quartic equations, provided the coefficient of the leading term is non-zero. However, since the general method is quite complex and susceptible to errors in execution, it is better to apply one of the special cases listed below ...
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A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive th root of unity, with an odd prime number. The uniqueness is a consequence of Galois theory , there being a unique subgroup of index 2 {\displaystyle 2} in the Galois group over Q ...
If d ≠ 1 then the Galois closure N of K contains a unique quadratic field k whose discriminant is d (in the case d = 1, the subfield Q is sometimes considered as the "degenerate" quadratic field of discriminant 1). The conductor of N over k is f, and f 2 is the relative discriminant of N over K. The discriminant of N is d 3 f 4. [6] [7]