Group Theory And Factor Groups?

It seems like you're a bit confused about what cosets are in general, because you say you know the elements of the factor group are the cosets but you want to know what they look like.

The thing to keep in mind is that we're regarding all the things in the "denominator" group as equivalent.

I'll go through the examples you mentioned, hopefully that will give you a better understanding.

R/Z: We're regarding all integers as basically the same. What's important is how any real number differs from being an integer, i.e. the fractional part.

For an arbitrary real number a, a + Z = {r in R: r = a + z for some z in Z}. That's just the definition of the coset, of course.

So what does that set look like? It's the set of all numbers that differ from a by an integer. For example, the coset 3.554 + Z is {0.554, 1.554, 2.554, ..., -0.446, -1.446, ...}: the set of all real numbers that are exactly 0.554 more than an integer (or 3.554 more than an integer). There's a set like this for every number in the interval [0, 1), containing all the real numbers that are just that much more than an integer.

In ring theory, we're talking about two-sided ideals rather than normal subgroups, and residue classes rather than cosets, but it's the same principle. The main thing to note is that we're adding elements to get the residue classes, not multiplying.

In the case R[x] / (x² + 1), the ideal (x² + 1) is all the polynomials that are multiples of x² + 1, i.e. that have x² + 1 as a factor. So we will consider two polynomials equivalent if you can get one from the other by adding some multiple of x² + 1.

So the equivalence class of x³ + 2x² + 4x - 1, for instance, contains all the following polynomials:

x³ + 4x - 3 (by subtracting 2 (x² + 1))

2x² + 3x - 1 (the original polynomial minus x (x² + 1))

3x - 3 (the last polynomial minus 2 (x² + 1))

x^4 + x³ + 4x - 4 (the original polynomial plus (x² - 3) (x² + 1))

In short, it can be written as {x³ + 2x² + 4x - 1 + p(x) (x² + 1): p(x) in R[x]}.

So in this quotient ring, how do we tell if two elements are equivalent and what does the quotient ring structure look like?

Well, suppose we have two polynomials p(x) and q(x). We can divide them both by x² + 1:

p(x) = r(x) (x² + 1) + s(x)

q(x) = t(x) (x² + 1) + u(x)

where s(x) and u(x) each have degree less than the degree of x² + 1, i.e. degree 0 or 1.

Then p(x) is in the same residue class as s(x) and q(x) is in the same residue class as u(x). But these can only be the same if s(x) = u(x) (because to be in the same residue class they have to differ by a multiple of x² + 1 and that multiple would have to be 0 since both s(x) and u(x) have order no more than 1). So what's important is the remainders, s(x) and u(x). Each distinct such remainder forms a unique residue class.

So the residue classes can be uniquely represented by the polynomials with degree 0 or 1, i.e. the polynomials of form a + bx where a and b are arbitrary real numbers. Each residue class looks like

{a + bx + (x² + 1) p(x): a, b in R, p(x) in R[x]}.

When we multiply two such elements, we get

(a + bx) (c + dx) = ac + (bc + ad) x + (bd)x²

= (ac - bd) + (bc + ad) x + (bd) (x² + 1)

hence the product is in the residue class represented by (ac - bd) + (bc + ad)x.

The fact that we're removing all multiples of x² + 1 means that every time we generate an x² term, it is equivalent to generating a -1 term. That's why this ring is isomorphic to the complex numbers, because by identifying the polynomial x² + 1 with 0, we're identifying x² with -1. Compare the multiplication of complex numbers: (a + bi) (c + di) = ac + (ad + bc)i + (bd) i² = (ac - bd) + (ad + bc) i.