Please explain this question to me (F2 field)

[Suy]

Please explain these question to me ...

Homework Statement

1)A system of m equations over F₂ in n
unknowns has exactly 2^n−m solutions.
• A. True
• B. False
answer: false

2)How many roots x \in F₂ does the polynomial
x2+x+1 have?
answer:0

3)Which of the following is a cube root of 8i? (Choose all correct answer)
A. sqrt(3)−i
B. −sqrt(3)−i
C. sqrt(3)+i
D. −2i
E. −sqrt(3)+i
answer: sqrt(3)+i( i think there is more?)

Homework Equations

The Attempt at a Solution

I have the answer, but I don't understand
also, those stuff is not in my textbook and the prof didn't give much example
... please help

 

[Office_Shredder]

Let's start the second and third questions, since they're more straightforward. For b, there are only two possible roots that you can check don't work. Do you know how to do this?

For c, can you just check whether each option is a cube root of 8i directly?

 

[Suy]

"b" you mean question 2?
No ,I have no clue... I don't know to cube root i...
The third question answer is come from wolfram..
By checking, Do you mean i cube every answer to see if it is equal 8i?
actually,I don't understand what's F2 is ...

 

[Office_Shredder]

Right, by b I meant the second question and c the third question

F₂ is just the field of two elements. It's the same thing as \mathbb{Z}/2\mathbb{Z}, the integers modulo 2.

If x is a cube root of 8i, then x³=8i. So can you check whether the five options given are cube roots of 8i?

 

[Suy]

answer is C,D,E, but is there a easy way to do it??
for question 2, I know there is no solution, but the question say x \in F₂ confused me
And what about question 1 please help me

edit:

If z/2z, is that mean (x+iy)/(2(x+iy))?
Thank you so much!