Proving b = a^{-1} = a^2 = a + 1 in Field F_4 with 1 + 1 = 0

[JG89]

Homework Statement

Let F_4 = {0,1,a,b} be a field containing four elements. Assume that 1 + 1 = 0. Prove that b = a^{-1} = a^2 = a + 1.

Homework Equations

The Attempt at a Solution

Do I assume that 0, 1, a, and b are al distinct elements? If so, then I can prove the question.

 

[VietDao29]

Homework Statement

Let F_4 = {0,1,a,b} be a field containing four elements. Assume that 1 + 1 = 0. Prove that b = a^{-1} = a^2 = a + 1.

Homework Equations

The Attempt at a Solution

Do I assume that 0, 1, a, and b are al distinct elements? If so, then I can prove the question.

Yes, they are all distinct elements.

Let F₄ = {0,1,a,b} be a field containing four elements.

That means, the 4 elements 0, 1, a, and b of the field F₄ are of course, distinct. :)