query: Given a field, the field of algebraic elements over it in an algebraically closed field extension is again algebraically closed?
positive_passages.docid: 19511
positive_passages.text: Is the sub-field of algebraic elements of a field extension of $K$ containing roots of polynomials over $K$ algebraically closed?
negative_passages.docid: 139499
negative_passages.text: The algebraic closure of a finite field and its Galois group
negative_passages.docid: 135611
negative_passages.text: Extensions of $\mathbb{Z}$ by $\mathbb{Z}$
negative_passages.docid: 103948
negative_passages.text: Minimal polynomial is invariant under field extensions
negative_passages.docid: 78412
negative_passages.text: Representing a countable field by $\Bbb N$ [transport of algebraic structure]
negative_passages.docid: 142336
negative_passages.text: Why does the integral domain "being trapped between a finite field extension" implies that it is a field?
negative_passages.docid: 51770
negative_passages.text: Number of elements in a finite field extension for finite fields
negative_passages.docid: 153589
negative_passages.text: Integral basis of an extension of number fields
negative_passages.docid: 111347
negative_passages.text: Finding an example of field extensions satisfying certain properties
negative_passages.docid: 146446
negative_passages.text: Infinite Degree Algebraic Field Extensions
negative_passages.docid: 97004
negative_passages.text: Algebraic closure not even finitely generated over base field
negative_passages.docid: 111444
negative_passages.text: Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?
negative_passages.docid: 64068
negative_passages.text: Is every finite Galois extension of $\mathbb{Q}_p$ the completion of a number field at some valuation?
negative_passages.docid: 68790
negative_passages.text: Do we have $K(\beta)=K(\beta^2)$ for field extension of odd degree?
negative_passages.docid: 157420
negative_passages.text: Question about Subrings and integraly closed
negative_passages.docid: 26999
negative_passages.text: How is the degree of the minimal polynomial related to the degree of a field extension?
negative_passages.docid: 69935
negative_passages.text: Field extensions of finite degree and primitive elements