Proposition: let k be a field and K it’s field of algebraic elements (textbook went through the proof essentially k[x]/k algebraic iff x is algebraic iff extension is finite. Since k[x][y]=k[x,y] and the vector space formula, k[x,y] is finite thus algebraic and the result follows). Then K is the algebraic closure of k. Proof: let P be any polynomial in K[X], a any root of P. We know that K[a]/K and K/k are algebraic. Then K[a]/k is algebraic that is a is algebraic over k and in K. So is this a generalization of the result in the textbook? And is the converse true? If a field k is algebraically closed, is it the algebraic closure of some field? And are all algebraic closures the set of algebraic elements of some field? The last one is true I think. The algebraic closure of a field is equivalent with the set of algebraic elements then? Something must be wrong here because they are not introduced in the same way.