Thursday, June 6, 2019

ring theory - Show that E1 is algebraic over K.




I found a problem in proving a theorem in field theory which states that Every field has an algebraic closure. Here's the sketch of the proof in the lecture notes given by our instructor.



Let K be a field. I want to find out the algebraic closure of K. For that we consider indeterminates Xf for each irreducible polynomial fK[X]. Now let us consider the polynomial ring R=K[Xf | fIrr(K[X])]. Let I be an ideal of R defined as I=f(Xf) | fIrr(K[X]). Then I is a proper ideal of R and hence by Krull's lemma I is contained in a maximal ideal say M. Let E1=R/M. Then E1 is an algebraic extension of K such that every irreducible polynomial in K[X] has a zero in E1. Continuing the same argument as before we get a chain of fields E0E1E2E3 with the property that every irreducible polynomial in Ei[X] has a zero in Ei+1 and Ei+1 is an algebraic field extension of Ei for each i=0,1,2,. (where K=E0). Take E=i0Ei. Then any irreducible polynomial in E[X] has a zero in E and since each Ei is an algebraic field extension of K so is E. Hence E is an algebraic closure of K.



In this sketch of the proof I understood everything except the part algebraic extension. I don't understand why Ei+1 is an algebraic extension of Ei. I observed that it is enough to show that E1 is an algebraic extension of K. But why is it so?



Can anybody please help me in this regard? Any suggestion will be highly appreciated.



Thank you very much for your valuable time.


Answer




For each irreducible fK[X], note that ¯Xf:=Xf+MR/M is algebraic over K, since f(Xf)M.



Now R/M is generated (as a K-algebra) by all the ¯Xf, i.e. each element is a sum of products of elements of K and elements of the form ¯Xf, each of which is algebraic over K. So, every element of R/M is algebraic, since sums and products of algebraic elements are algebraic.


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