# Solution of an equation in a certain field

Let $F$ be a certain field.

Prove or disprove that the following statements:

• The equation $X^3=0_F$ has only one solution.

• The equation $X^3=1_F$ has only one solution.

• Suppose F is finite, then the equation $X^3=1_F$ has only one solution.

I'm pretty sure all of them are true but I'm not sure how to write the proof so any help would be appreciated.

Note: I'm only in my first few weeks of linear algebra so I don't think any advanced solutions would help.

$X^3 = 0$ has only one solution. This is because fields do not have zero divisors, which follows from requiring every element to be invertible.
$X^3 = 1$ might have more solutions. In $\mathbb{C}$, for example, there are three cube roots of unity.
For finite fields we can, again, have more than two solutions. Consider $\mathbb{F}_7$ - here both 1 and 2 are solutions, because $2^3 = 8 = 1 \;mod\; 7$.