# 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 Answer

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 $\mathbf Q$?

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About 2: an algebraically closed field is infinite and a field of characteristic 0 is infinite. It is not that weird to ask wether there is an inclusion. –  Siméon Jan 14 '13 at 17:22
@Ju'x Ah thanks, that makes sense. –  jathd Jan 14 '13 at 17:58
@Ju'x: inclusion is a homomorphism (injection), ie trivial kernel, ie characteristic is preserved. –  Mark Bennet Jan 14 '13 at 19:49