RNT1.4.1. Example of Quotient Ring

Abstract Algebra: Are there fields F such that the rings F[x]/(x^2) and F[x]/(x^2-1) are isomorphic? We construct an isomorphism when char F = 2.

[MathDoctorBob]

Thanks! Rings take more practice than groups, so keep at it.

1) In practice, yes. But you should keep the ideals around until you're convinced they don't matter notationally.

2) Using the homomorphism property: f(1) = 1, so
f(a+bx) = f(a) + f(bx) = a + bf(x) = a + b(1+y) = (a+b) + by

[MathDoctorBob]

Left as an exercise. You can invert f using g(c+dy) = (c-d) + dy. I'll annotate.

Or show directly. For instance, f(a+by) = 0 means (a+b) + by =k(y)y^2 or a=b=0. So one-one.

[MathDoctorBob]

All we need is that characteristic equal 2. So Z/2 works, but you can also use any field with 2^n elements. Also rational functions over Z/2, Laurent series over Z/2, ....

[mcsquared]

I love you Dr BOB! Thanx for the math.

[MathDoctorBob]

Thanks! I'm a big fan of these. - Bob

[surferdudevideo]

Great video, I'm not sure if I completely understand though and here are a few questions.

1) Think x^2 = 0 and y^2=1 is there just to simplify the notation, correct?

2) How is saying f(x)= 1+y the same as saying f(a+bx)=(a+b)+by ?

Thanks!

[goodboyelvis]

I don't understand how bijection was achieved. Can you explain why we don't show injection and surjection here?