Unformatted text preview: by Proposition 1.6.2 . We prove existence by means of a generalization of the matrix formulation of the Euclidean algorithm. Lemma 1.6.20. Given nonzero integers a 1 ;a 2 ;:::;a n ( n ² 2 ), there is a natural number d and an n –by– n integer matrix Q such that Q is invert-ible, Q ± 1 also has integer entries, and .d;0;:::;0/ D .a 1 ;a 2 ;:::;a n /Q: Proof. We proceed by induction on n . The base case n D 2 is established in the proof of Proposition 1.6.10 . Fix n > 2 and suppose the assertion holds for lists of fewer than n nonzero integers. Then there exists a natural number d 1 and a n ± 1 –by– n ± 1 integer matrix Q 1 with integer inverse such that .d 1 ;0;:::;0/ D .a 2 ;:::;a n /Q 1 : By the base case n D 2 , there is a natural number d and a 2 –by– 2 integer matrix Q 2 with integer inverse such that .d;0/ D .a 1 ;d 1 /Q 2 :...
View Full Document
- Fall '08
- Algebra, Natural number, Prime number, Greatest common divisor, Euclidean algorithm