This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 abi ). 7. The following questions concern R/ 5 R . (a) What is the factorization of x 241 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 ....
View
Full
Document
 Spring '09
 BRUBAKER
 Algebra, Number Theory, Integers

Click to edit the document details