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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".
Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for higher-degree number fields. For example, there are two non-isomorphic cubic fields of discriminant 3969. They are obtained by adjoining a root of the polynomial x 3 − 21x + 28 or x 3 − 21x − 35, respectively.
Example: the discriminant of a binary quadratic form. These symbols can be explained by the following example from Gordan. [5] Suppose that () = + + is a binary ...
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.)
A form of discriminant is definite if <, degenerate if is a perfect square, and indefinite otherwise. A form is primitive if its content is 1, that is, if its coefficients are coprime. If a form's discriminant is a fundamental discriminant, then the form is primitive. [1]
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 ...
The discriminant of a quadratic form Q 3 is = ... An example of Bhargava cube. The three quadratic forms associated with the numerical Bhargava cube shown in the ...