Friday, 26 April 2019

abstract algebra - Example check: two algebraically closed fields with one a subset of the other




The problem asks to find two different algebraically closed fields E and F with EF.



We have not done a whole lot of stuff with algebraically closed and in fact the only one that came to mind was C.



Does C(x), the field of rational functions in x with coefficients from C work? Our definition of an algebraically closed field (which I'm not sure if there are other standard definitions or not) is that every polynomial with coefficients from the field must have a root in the field. Since the set of polynomials in C(x) is just C[x] it seems like my example is algebraically closed as well. Does this in fact work or am I missing something?



Thanks




Edit: I did notice a flaw in my thinking. I need to think of polynomials with coefficients in C(x) not polynomials contained in C(x).




Answer



The algebraic closure of Q,¯QC gives an example to your main question.



More generally, for every (edit : u uncountable) cardinality, there is a unique algebraically closed field (upto isomorphism) So while C¯C(X)(the algebraic closure of C(X)) are another pair of examples, the two fields are isomorphic although they are not equal.



(The uniqueness requires the axiom of choice.)


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