query: A field extension that is a finitely generated algebra is an algebraic extension
positive_passages.docid: 20218
positive_passages.text: Is this field extension finite?
negative_passages.docid: 6824
negative_passages.text: Extensions of degree two are Galois Extensions.
negative_passages.docid: 140938
negative_passages.text: Finitely generated projective module
negative_passages.docid: 93111
negative_passages.text: Finiteness of the Algebraic Closure
negative_passages.docid: 139499
negative_passages.text: The algebraic closure of a finite field and its Galois group
negative_passages.docid: 144211
negative_passages.text: a second course in abstract algebra
negative_passages.docid: 16589
negative_passages.text: Field of fractions of integral extension is an algebraic extension
negative_passages.docid: 49521
negative_passages.text: In an extension of finitely generated $k$-algebras the contraction of a maximal ideal is also maximal
negative_passages.docid: 19525
negative_passages.text: Showing field extension $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})/\mathbb{Q}$ degree 8
negative_passages.docid: 24168
negative_passages.text: A characterisation of quadratic extensions contained in cyclic extensions of degree 4
negative_passages.docid: 17378
negative_passages.text: Characterize finite dimensional algebras without nilpotent elements
negative_passages.docid: 115129
negative_passages.text: Constructing an explicit isomorphism between finite extensions of finite fields
negative_passages.docid: 38028
negative_passages.text: finitely generated k-algebra and polynomial ring
negative_passages.docid: 134896
negative_passages.text: Enlightening proof that the algebraic numbers form a field
negative_passages.docid: 127029
negative_passages.text: Ring of formal power series finitely generated as algebra?
negative_passages.docid: 103654
negative_passages.text: A finite field extension that is not simple
negative_passages.docid: 79450
negative_passages.text: For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$