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. :)

 

[Architeuthis Dux]

Yes, in this case, 0, 1, a, and b are all distinct elements in the field F_4. In order to prove that b = a^{-1} = a^2 = a + 1, we can use the fact that 1 + 1 = 0 in this field.

First, we can show that b = a^{-1} by using the definition of inverse elements in a field. We know that for any element a in a field, there exists an element a^{-1} such that a * a^{-1} = 1. In this case, we can see that a * b = 1, since 1 + 1 = 0. Therefore, b = a^{-1}.

Next, we can show that a^2 = a + 1 by using the fact that 1 + 1 = 0. We can rewrite this as 1 = -1, and then use this to simplify a^2 as follows:

a^2 = a * a = a * (1 + 1) = a * (-1) = -a = a + 1.

Finally, we can show that a + 1 = b by using the fact that 1 + 1 = 0. We can rewrite this as 1 = -1, and then use this to simplify a + 1 as follows:

a + 1 = a + (1 + 1) = a + (-1) = a * (-1) = -a = b.

Therefore, we have shown that b = a^{-1} = a^2 = a + 1, using the fact that 1 + 1 = 0 in the field F_4. This proves the statement provided in the homework.