Let W= -1+-3 2 (a) (1 point) Prove that 1 - w is a prime number...

  

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Let W= -1+√-3 2 (a) (1 point) Prove that 1 - w is a prime number in the Eisenstein Integers. (b) (3 points) The fundamental parallelogram associated to a Eisenstein integer z is the parallelo- gram with corners 0, z, wz and (1+w)z with the segments with (1+w) z as a vertex removed. Using the integers in the fundamental parallelogram associated to (1 − w)², find the addition and multiplication tables of the Eisenstein Integers modulo (1 − w)². (c) (1 point) Using the previous table, verify that the Eisenstein Integers modulo (1-w)² are the same as the rational integers modulo 9. (d) (2 points) Let N be an integer. A cubic residue modulo N is an invertible class A modulo N such that the congruence X³ = A (mod N) is solvable. Find all cubic residues' (e) (3 points) Let X, Y, Z be nonzero Eisenstein Integers relatively prime to 1 - w. Is it possible to have X³ + Y³ + Z³ = 0? Let W= -1+√-3 2 (a) (1 point) Prove that 1 - w is a prime number in the Eisenstein Integers. (b) (3 points) The fundamental parallelogram associated to a Eisenstein integer z is the parallelo- gram with corners 0, z, wz and (1+w)z with the segments with (1+w) z as a vertex removed. Using the integers in the fundamental parallelogram associated to (1 − w)², find the addition and multiplication tables of the Eisenstein Integers modulo (1 − w)². (c) (1 point) Using the previous table, verify that the Eisenstein Integers modulo (1-w)² are the same as the rational integers modulo 9. (d) (2 points) Let N be an integer. A cubic residue modulo N is an invertible class A modulo N such that the congruence X³ = A (mod N) is solvable. Find all cubic residues' (e) (3 points) Let X, Y, Z be nonzero Eisenstein Integers relatively prime to 1 - w. Is it possible to have X³ + Y³ + Z³ = 0?