781hw8 - (mod 3 4 A prime π ∈ R is called primary if π...

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18.781, Fall 2007 Problem Set 8 Due: FRIDAY, November 2 These exercises continue to develop the theory of algebraic integers needed for cubic reciprocity. In all the following problems, let R = Z [ ω ], with ω = - 1+ i 3 2 . 1. Prove that the number of residue classes in R/πR is N ( π ). (That is, rewrite the portion of the proof done in class, and then finish the proof by showing the representatives we chose are indeed distinct mod π .) 2. Factor the following elements of R into primes (in R , of course): 7, 21, 45, 22, 143 (and prove that your factors are indeed prime). 3. As we’ll discuss in lecture on Monday, it is often convenient to choose a particular representative from the set of elements defined up to a choice of units. For rational numbers, we chose n from the set { n, - n } . For an element η R , we must choose from among { η, - η,ωη, - ωη,ω 2 η, - ω 2 η } . Prove that for any prime π with N ( π ) = p 1 (3) , exactly one of these six elements (i.e. π multiplied by some unit in R ) is equivalent to 2
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Unformatted text preview: (mod 3). 4. A prime π ∈ R is called primary if π ≡ 2 (3). Factor 19 in R , and find primary primes which are “associates” of each prime factor (that is, they differ from the prime factor by a multiple of a unit). 5. Prove that primitive roots exist for R/πR , where π is a prime in R . Conclude that ± α π ² 3 = 1 if and only if α is a cubic residue mod π. Recall that the symbol is defined by the congruence ± α π ² 3 ≡ α N ( π )-1 / 3 ( π ) . 6. Show that ± α π ² 3 = ± α π ² 2 3 = ³ α 2 π ´ 3 = ± ¯ α ¯ π ² 3 , where ¯ α denotes the complex conjugate ( a + bi 7→ a-bi ). 7. The following questions concern R/ 5 R . (a) What is the factorization of x 24-1 in R/ 5 R ? (b) How many cubic residues are there in R/ 5 R ? (c) Show that ω (1-ω ) has order 8 in R/ 5 R and that ω 2 (1-ω ) has order 24 in R/ 5 R ....
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781hw8 - (mod 3 4 A prime π ∈ R is called primary if π...

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