Sunday, October 21, 2018

abstract algebra - Algebraic Closure & Algebraic Elements

I learn that the algebraic closure of rational numbers are algebraic numbers. Is this true in general, namely




For all field $F$ and its algebraic closure $\bar F$, does $\alpha\in \bar F$ imply $\alpha$ is algebraic over $F$?




My idea:



The set of algebraic elements in the algebraic closure $F$ forms a field, since for every algebraic elements $\alpha, \beta \in \bar F$, $\alpha+\beta, -\alpha, \alpha \beta, \alpha^{-1} \in F(\alpha, \beta)$. But $[F(\alpha, \beta)]= [F(\alpha)(\beta): F(\alpha)][F(\alpha):F]$ is finite, so all the four of them is again an algebraic element.




I think that what I conjectured is not true. Actually, in the famous proof of the existence of algebraic closure over a field (due to E. Artin), the algebraic closure of $F$ is constructed as follows:




  1. Construct a field $F_1$ over $F_0:=F$ by taking quotient ring over a huge polynomial ring and its maximal ideal so that every polynomial $p(x)\in F[x]$ has a root.

  2. Construct $F_2$, $F_3$, ... similarly.

  3. Define $E=\bigcup_{k=0}^{\infty} F_k$. Then $E$ is the algebraic closure of $F$ that we want.



If the field of all algebraic numbers over any field is algebraic closed, then we do not have to construct $F_2$, $F_3$, ... sequentially. It seems that if $F$ is perfect, then every algebraic element over $F$ is contained in $F_1$, so $F_1$ is already sufficient. We don’t have to bother to write down the rest of the proof.




Of course, this is not a proof, but I can’t find any counterexample. Any supporting materials can be helpful. Thanks in advance.






As Don Thousand said in the comments, this is true since it is impossible to add transcendental elements in the process I mentioned. However, I cannot come up with a proof that explains why elements in $F_2$, $F_3$, ... remains to be algebraic, nor did I find any material that support that claim. Is that true? Can you please provide a prove?

No comments:

Post a Comment

analysis - Injection, making bijection

I have injection $f \colon A \rightarrow B$ and I want to get bijection. Can I just resting codomain to $f(A)$? I know that every function i...