HW solutions 9

HW solutions 9 - Math 3320 Problem Set 9 Solutions 1 1....

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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 coeFcients 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 coeFcients 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. Solution : If the values of the form Q(x,y) = ax 2 + bxy + cy 2 are all divisible by d then in particular the values Q( 1 , 0 ) = a and Q( 0 , 1 ) = c are divisible by d . Also the value Q( 1 , 1 ) = a + b + c is divisible by d , but since a and c are divisible by d , this implies that b is divisible by d . Thus we have Q(x,y) = d( a d x 2 + b d xy + c d y 2 ) = dQ (x,y) . (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. Solution : This statement is logically equivalent to the statement that a discriminant is not fundamental if and only if it is equal to a square times the discriminant of some other form. In one direction: If a discriminant Δ is not fundamental then there exists a nonprimi- tive form of discriminant Δ . This form Q can be written as dQ for some other form Q (where d > 1), hence Δ = d 2 Δ where Δ is the discriminant of Q . Thus Δ is a square times the discriminant of some other form. In the other direction: If Δ is a square d 2 times the discriminant Δ of some form Q , then the form dQ is a nonprimitive form of discriminant Δ , hence Δ is not a fundamental discriminant. (c) Make a list of all the discriminants between 50 and + 50 that are fundamental and another list for those that are not fundamental. Solution : ±irst, here are all the discriminants between 50 and 50: 48 , 47 , 44, 43 , 40 , 39 , 36 , 35 , 32 , 31 , 28 , 27 , 24 , 23 , 20 , 19 , 16 , 15, 12 , 11 , 8 , 7 , 4 , 3 , 0 , 1 , 4 , 5 , 8 , 9 , 12 , 13 , 16 , 17 , 20 , 21 , 24 , 25 , 28 , 29, 32 , 33 , 36 , 37 , 40 , 41 , 44 , 45 , 48 , 49. The ones that are a square times an-
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This note was uploaded on 04/14/2010 for the course MATH 3320 at Cornell.

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HW solutions 9 - Math 3320 Problem Set 9 Solutions 1 1....

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