Show that Z[i] /⟨1-i⟩is a field. How many elements does this field have?

Name: Problem 58, Chapter 14: Contemporary Abstract Algebra
Uploaded: 2021-03-25T21:19:36-08:00
Duration: 6 min 2 s
Channel: Chris Trentman
Description: Show that Z[i] /⟨1-i⟩is a field. How many elements does this field have?

step 1 We're asked to prove statements about arbitrary complex numbers. So let Z and W be some complex

Show that Z[i] /⟨1-i⟩is a field. How many elements does this...

Key Concepts

Norm on Gaussian Integers

The norm of a Gaussian integer, defined by N(a + bi) = a² + b², serves as a useful tool in analyzing divisibility and factorization properties within Z[i]. In the context of quotient rings, the norm can also help in determining the number of elements in the field by relating to the size of the ideal's generated residue class.

Quotient Rings

A quotient ring, formed by partitioning a ring using an ideal, is the set of cosets. The structure of the quotient ring reflects in many cases the properties of the ideal used. When the ideal is maximal, the resulting quotient ring is a field, which means every non-zero element has a multiplicative inverse.

Gaussian Integers

The Gaussian integers, denoted by Z[i], are complex numbers of the form a + bi, where both a and b are integers. They form a ring that extends the usual integers, allowing for a richer structure for exploring divisibility, factorization, and ideal theory within an algebraic number theoretical context.

Ideals and Maximal Ideals

An ideal in a ring is a subset that absorbs multiplication by elements of the ring and can be used to form quotient rings. A maximal ideal is one that is as large as possible without being the whole ring; in a commutative ring with unity, the quotient by a maximal ideal results in a field. Establishing that an ideal is maximal is a standard way to prove that the corresponding quotient ring possesses a field structure.

Transcript

00:01 We're asked to prove statements about arbitrary complex numbers.

00:06 So let z and w be some complex numbers.

00:13 In part a, we're asked to prove that the real part of z is one -half of z plus z conjugate.

00:24 Now, to prove this statement, let's actually avoid decomposing a z, or if we're going to decompose z, recognize that z is equal to the real part of z plus the imaginary part of z times i and then from this perspective it follows that z conjugate is equal to the real part of z and we flip the sign of the imaginary parts this is minus the imaginary part of z times i and therefore z plus z conjugate will we cancel out the imaginary part of z term so we just get two times the real part of z.

01:19 Solving for the real part of z, it's clear that the real part of z is equal to one half of z plus z conjugate.

01:29 This is what we wanted to prove in part a.

01:33 Ok, and in part b, we're likewise asked to show that the imaginary part of z is 1 half of z minus z conjugate.

01:43 So once again, we have z and z conjugate above, but now z, z might as the middle.

01:49 Minus z conjugate, well now the real part drops out and we have two times the imaginary part of z times i.

02:00 And so to solve for the imaginary part of z, i'm going to divide both sides by 2i.

02:09 So we get that the imaginary part of z is equal to 1 over 2i times z minus z conjugate.

02:17 Now let's simplify 1 over 2i.

02:39 Well, 1 over 2i, multiply the top and the bottom by the complex conjugate, which is negative 2i.

02:47 We get negative 2i over negative 4 i squared.

02:53 This is simply negative 2i over 4, which is negative 1⁄2i.

03:02 And so this is equal to negative 1⁄2i times 0 .5i.

03:11 Z minus z conjugate.

03:15 Yeah, i don't think the book was actually right on this one.

03:19 The correct answer should really get the imaginary part of z is 1 over 2i times z minus z conjugate...

Please give Ace some feedback