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Unformatted text preview: (mod 3). 4. A prime R is called primary if 2 (3). Factor 19 in R , and nd primary primes which are associates of each prime factor (that is, they dier from the prime factor by a multiple of a unit). 5. Prove that primitive roots exist for R/R , where is a prime in R . Conclude that 3 = 1 if and only if is a cubic residue mod . Recall that the symbol is dened by the congruence 3 N ( )-1 / 3 ( ) . 6. Show that 3 = 2 3 = 2 3 = 3 , where 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 (1- ) has order 8 in R/ 5 R and that 2 (1- ) has order 24 in R/ 5 R ....
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