# A multiple choice question on field extensions

Which of the following statement(s) is / are true?
1. $\mathbb{C} (x)$ is algebraically closed, where $x$ is an indeterminate.
2. An algebraically closed field must be of characteristic $0$.
3. If $E$ is an algebraically closed extension field of $F$, then $E$ is an algebraic extension of $F$.

can anyone help me to solve this problem.thanks for your time.

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What have you tried? –  Chris Eagle Jan 14 '13 at 17:15

1. The question is whether all polynomials with coefficients in $\mathbf C(x)$ have a root. Hint: consider $f(T) = T^2-x$. In other words, does $x$ have a square root in $\mathbf C(x)$?
2. That's a bit weird; why would that be true? Anyway, if you are familiar with the fact that every field has an algebraic closure, look at the algebraic closure of $\mathbf F_2$. What can you say about its characteristic?
3. What do you think of $\mathbf C$ over $\mathbf Q$?