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# LS11-12 - S UMMARY OF 11/12 L ECTURE We began class with a...

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UMMARY OF 11/12 L ECTURE We began class with a quick refresher on the geometry of the complex plane, C . Most im- portantly, we discussed the magnitude of a complex number: it is the physical distance from the point representing that number, to the origin. Thus, | i | = 1 , despite the temptation to say that the ‘distance’ from i to 0 is i . It is easy to prove (by Pythagorean theorem!) that | a + bi | = a 2 + b 2 . We then returned to our theorem that Z × p is cyclic. Recall that last time we showed that any ﬁnite nice group G must be cyclic – here ‘nice’ means that for every n 1 , x n = 1 has at most n distinct solutions in G . Thus, to prove our theorem it remains only to show that Z × p is nice. We will prove rather more: Theorem 1. Suppose f ( x ) is a monic polynomial with integer coefﬁcients, and that deg f 1 . Then the number of distinct elements x Z p satisfying f ( x ) 0 ( mod p ) is at most the degree of f . [Recall that f has degree n (written deg f = n ) if the highest power of x appearing in f ( x ) is x n . f is monic if the coefﬁcient of the highest-degree term is 1. Thus a monic polynomial of degree n looks like f ( x ) = x n + c 1 x n - 1 + c 2 x n - 2 + ··· + c n - 1 x + c n .] It will be convenient to deﬁne Z [ x ] to be the collection of all polynomials with integer coefﬁ- cients. Proof. Let Z p ( f ) = { x Z p : f ( x ) 0 ( mod p ) } . We need to show that | Z p ( f ) | ≤ deg f for every monic polynomial f Z [ x ] with deg f 1 . We do this by induction: deg f = 1 : In this case, we must have f ( x ) = x - a for some integer a . It is a good exercise to show that

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LS11-12 - S UMMARY OF 11/12 L ECTURE We began class with a...

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