# Please explain this question to me (F2 field)

1. Sep 25, 2010

### Suy

Please explain these question to me ....

1. The problem statement, all variables and given/known data

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

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

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?)

2. Relevant equations

3. 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

Last edited: Sep 25, 2010
2. Sep 25, 2010

### Office_Shredder

Staff Emeritus
Re: Please explain this question to me .... (F2 field)

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?

3. Sep 25, 2010

### Suy

Re: Please explain this question to me .... (F2 field)

"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 ...

Last edited: Sep 25, 2010
4. Sep 25, 2010

### Office_Shredder

Staff Emeritus
Re: Please explain this question to me .... (F2 field)

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?

5. Sep 25, 2010

### Suy

Re: Please explain this question to me .... (F2 field)

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