1. We will be working in the ring Z[5] this question with norm N(a+b-5):=a + 5...

Transcribed Image Text:

1. We will be working in the ring Z[√5] this question with norm N(a+b√-5):=a² + 5 xb². (a) Find all elements with norm less than 4 in Z[√-5], and explain why this implies there are no universal side divisors in Z[√-5]. Identify two elements c and d (which are different from those chosen by anyone else in the class and do not have a or b as zero) such that N(c) > N(d) and god (N(c), N (d)) is equal to 2 or 3. Find the quotient that gives the remainder with smallest norm when you divide c by d. [4] (b) Let J be the non-principal ideal generated by both c and d, carefully identify its lattice of points, showing how they result from c and/or d. Explain why J is not a principal ideal and identify the cosets in Z[√-5]/J. [3] 1. We will be working in the ring Z[√5] this question with norm N(a+b√-5):=a² + 5 xb². (a) Find all elements with norm less than 4 in Z[√-5], and explain why this implies there are no universal side divisors in Z[√-5]. Identify two elements c and d (which are different from those chosen by anyone else in the class and do not have a or b as zero) such that N(c) > N(d) and god (N(c), N (d)) is equal to 2 or 3. Find the quotient that gives the remainder with smallest norm when you divide c by d. [4] (b) Let J be the non-principal ideal generated by both c and d, carefully identify its lattice of points, showing how they result from c and/or d. Explain why J is not a principal ideal and identify the cosets in Z[√-5]/J. [3]

Answer rating: 100% (QA)

a To find all elements with norm less than 4 in Z5 we need to consider all possible combinations of a and b that satisfy the equation Naby5 a5b 4 Sinc... View the full answer