Question
Can anyone help me with my math project?
Thanks!
The basic notions of primality and irreducibility behave a little differently in other rings. This project will go through a specific example of the gaussian integers, and the way a few theorems that work in the integers don't work there. The gaussian integers, denoted Z[i] are objects of the form (a, b) = a + bi where a. 6 z. The haw an addition given by (a + bi) + (c + di) = (a + c)+(b + d)i and a multiplication given by (a + bi)(c.+ di) = (ac - bd) + (bc + ad)i. The addition and multiplication are both commutative and associative and the distributative laws apply. These properties, when all taken together, make Z[i] into a commutative ring, much like the integers. Define |a + bi| =. Show that if a + bi 0 then |a + bi 1. Define a bi = a + (-b)i. Show that |a + bi| = Show that |(a + bi)(c + di)| = |(a + bi)\(c + di)| Define a unit as a number which has an inverse. Show that the only units in Z[i] are 1, - l.i, - i. Recall that irreducible means that if x = ab then either a or b is a unit. Show that 2 + i is irreducible. Recall that prime means that if xb then x|a or Show that 2 + i is irreducible.
Explanation / Answer
question.?
