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Then the discriminant is the product of the a i, which is well-defined as a class in K/(K ×) 2. 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 ...
The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871. [16] At this point, he already knew the relationship between the discriminant and ramification. [17] Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857. [18]
For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field () with x 3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place that does.
Explicitly it is the modular discriminant Δ ( z , q ) , {\displaystyle \Delta (z,q),} which represents (up to a normalizing constant ) the discriminant of the cubic on the right side of the Weierstrass equation of an elliptic curve ; and the 24-th power of the Dedekind eta function .
If only one root, say r 1, is real, then r 2 and r 3 are complex conjugates, which implies that r 2 – r 3 is a purely imaginary number, and thus that (r 2 – r 3) 2 is real and negative. On the other hand, r 1 – r 2 and r 1 – r 3 are complex conjugates, and their product is real and positive. [ 23 ]
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the discriminant can be shown to be (up to sign) p p−2, by linear algebra methods. More precisely, only totally ramified primes have a chance of being Eisenstein primes for the polynomial. (In quadratic fields, ramification is always total, so the distinction is not seen in the quadratic case like x 2 + x + 2 above.)
French police officers made two arrests after a brief fan-fight at a Nations League soccer game between France and Israel, Paris police chief Laurent Nunez said Friday. Stewards “immediately ...
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