HW solutions 8

HW solutions 8 - Math 3320 Problem Set 8 Solutions 1 1...

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Unformatted text preview: Math 3320 Problem Set 8 Solutions 1 1. Determine the number of equivalence classes of quadratic forms of discriminant Δ = 120 and list one form from each equivalence class. Solution : We use the relationship Δ = h 2 + 4 pq which holds for each edge of the separator line, where h is the label on the edge and the adjacent regions are labeled p and- q , with p > 0 and q > 0. We also know that h has the same parity as Δ , because of the formula Δ = h 2 + 4 pq . Now we make a table of all the possibilities for h , p , and q when Δ = 120: h pq (p,q) 30 ( 1 , 30 ), ( 2 , 15 ), ( 3 , 10 ), ( 5 , 6 ), ( 6 , 5 ), ( 10 , 3 ), ( 15 , 2 ), ( 30 , 1 ) 2 29 ( 1 , 29 ), ( 29 , 1 ) 4 26 ( 1 , 26 ), ( 2 , 13 ), ( 13 , 2 ), ( 26 , 1 ) 6 21 ( 1 , 21 ), ( 3 , 7 ), ( 7 , 3 ), ( 21 , 1 ) 8 14 ( 1 , 14 ), ( 2 , 7 ), ( 7 , 2 ), ( 14 , 1 ) 10 5 ( 1 , 5 ), ( 5 , 1 ) (Notice incidentally that the values for pq decrease regularly, by 1 , 3 , 5 , 7 , 9.) To find forms that realize all the possibilities in the table we start with the principal form x 2- 30 y 2 , whose separator line is shown in the left half of the following figure: This realizes all the pairs (p,q) in the table with p equal to 1 or 6. To realize ( 2 , 15 ) we need a new form, so we take 2 x 2- 15 y 2 with topograph shown in the right half of the figure above. This takes care of all the pairs (p,q) with p = 2 or p = 3. To get all the remaining pairs (p,q) we use the negatives of these two forms,- x 2 + 30 y 2 and- 2 x 2 + 15 y 2 . Thus all forms of discriminant 120 are equivalent to one of the four forms x 2- 30 y 2 , 2 x 2- 15 y 2 ,- x 2 + 30 y 2 and- 2 x 2 + 15 y 2 . No two of these four forms are equivalent to each other since their separator lines are different (the two separator lines shown and the negatives of these two separator lines). 2. Do the same thing for Δ = 61. Solution : This is similar, but now h is odd since Δ is odd. The table is: Math 3320 Problem Set 8 Solutions 2 h pq (p,q) 1 15 ( 1 , 15 ), ( 3 , 5 ), ( 5 , 3 ), ( 15 , 1 ) 3 13 ( 1 , 13 ), ( 13 , 1 ) 5 9 ( 1 , 9 ), ( 3 , 3 ), (...
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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 8 - Math 3320 Problem Set 8 Solutions 1 1...

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