3320hw9 - Math 3320 Problem Set 9 1 1 Preliminary comments...

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Unformatted text preview: Math 3320 Problem Set 9 1 1. Preliminary comments: A quadratic form Q(x,y) = ax 2 + bxy + cy 2 is called primitive if the greatest common divisor of the coefficients a,b,c is 1. If Q is not primitive, it can obviously be written as dQ ′ where Q ′ is primitive and d is the greatest common divisor of the coefficients of Q . The discriminant of Q is d 2 times the discriminant of Q ′ in this case. (a) Show that if all the values Q(x,y) of a quadratic form Q are multiples of some number d > 1 then Q = dQ ′ for some quadratic form Q ′ , hence Q is not primitive. (b) A discriminant is called fundamental if every quadratic form of that discriminant is primitive. Show that a discriminant is fundamental if and only if it is not equal to a square times the discriminant of some other form. (c) Make a list of all the discriminants between − 50 and + 50 that are fundamental and another list for those that are not fundamental....
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3320hw9 - Math 3320 Problem Set 9 1 1 Preliminary comments...

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