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A quadric is said to be non-degenerate if the matrix is invertible. A non-degenerate quadric is non-singular in the sense that its projective completion has no singular point (a cylinder is non-singular in the affine space, but it is a degenerate quadric that has a singular point at infinity).
In general a quadratic field of field discriminant can be obtained as a subfield of a cyclotomic field of -th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant, a special case of the conductor-discriminant formula.
A tennis court is the venue where the sport of tennis is played. It is a firm rectangular surface with a low net stretched across the centre. The same surface can be used to play both doubles and singles matches. A variety of surfaces can be used to create a tennis court, each with its own characteristics which affect the playing style of the game.
Simultaneously generalizing the case of imaginary quadratic fields and cyclotomic fields is the case of a CM field K, i.e. a totally imaginary quadratic extension of a totally real field. In 1974, Harold Stark conjectured that there are finitely many CM fields of class number 1. [12] He showed that there are finitely many of a fixed degree.
The two families of lines on a smooth (split) quadric surface. In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine ...
Geometrically, the discriminant of a quadratic form in three variables is the equation of a quadratic projective curve. The discriminant is zero if and only if the curve is decomposed in lines (possibly over an algebraically closed extension of the field). A quadratic form in four variables is the equation of a projective surface.
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Real tennis (also royal tennis or court tennis): An indoor racket sport which was the predecessor of the modern game of (lawn) tennis. The term real is used as a retronym to distinguish the ancient game from the modern game of lawn tennis. Known also as court tennis in the United States or royal tennis in Australia. [113]