In mathematics, particularly in algebra, the discriminant is a specific quantity associated with a polynomial equation that provides information about the nature of its roots. The most common context in which the discriminant is discussed is in quadratic equations, which are polynomial equations of the form: \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are coefficients, and \( a \neq 0 \).
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For a degree- polynomial over a field, with and roots in an algebraic closure, its discriminant isIt is a polynomial with integer coefficients in the coefficients of , and lies in the original coefficient field. Its vanishing is equivalent to a repeated root of a polynomial in an algebraic closure, even when the coefficient field itself contains no roots. This detects separability without solving the polynomial equation.[ref]
For a quadratic polynomial, the expression is the quadratic discriminant . For the cubic polynomial , it is . Thus has discriminant over , but after reduction to characteristic three its discriminant is zero and the resulting polynomial has a repeated root in its splitting field. Discriminants also indicate exceptional parameter values where distinct polynomial roots collide.