We prove that the types in Separably Closed Hasse Fields which have transcendence degree 1 are separably thin

We classify projective planes in algebraic combinatorial geometries in arbitrary fields of characteristic zero. We investigate the first-order theories of such geometries and pregeometries. Then we classify the algebraic combinatorial geometries of arbitrary field extensions of the transcendence degree ≥ 5 and describe their groups of automorphisms. Our results and proofs extend similar results...

art of Kant’s reason for writing the first Critique is to show the limits of theoretical reason; one of these limits is its inability to experience transcendence, to cognize supersensible objects, to intuit the sublime. But Kant doesn’t thereby want to make these experiences impossible—indeed, part of his reason for writing the second and third Critiques, along with such works as Religion withi...

Let K be a field, K[X] = K[X1, . . . , Xn] the polynomial ring in n variables over K for some n ∈ N, and K(X) the field of fractions of K[X]. Assume that L is a subfield of K(X) containing K. Then, the Fourteenth Problem of Hilbert asks whether the Ksubalgebra L ∩K[X] of K[X] is finitely generated. Zariski [17] showed in 1954 that the answer to this problem is affirmative if the transcendence d...

We investigate the amenability of skew field extensions of the complex numbers. We prove that all skew fields of finite Gelfand-Kirillov transcendence degree are amenable. However there are both amenable and non-amenable finitely generated skew fields of infinite Gelfand-Kirillov transcendence degree. AMS Subject Classifications: 12E15, 43A07

Let A be a finitely generated domain of GK dimension less than 3 over a field K and let Q(A) denote the quotient division algebra of A. Using the ideas of Smoktunowicz, we show that if D is a finitely generated division subalgebras of Q(A) of GK dimension at least 2, then Q(A) is finite dimensional as a left D-vector space. We use this to show that if A is a finitely generated domain of GK dime...

Given an affine domain of Gelfand–Kirillov dimension 2 over an algebraically closed field, it is shown that the centralizer of any non-scalar element of this domain is a commutative domain of Gelfand–Kirillov dimension 1 whenever the domain is not polynomial identity. It is shown that the maximal subfields of the quotient division ring of a finitely graded Goldie algebra of Gelfand– Kirillov di...

The authors conducted this study to further validate the revised short form of the Gratitude, Resentment, and Appreciation Test by investigating the relationship between GRAT-measured gratitude and two other constructs: (a) spiritual transcendence and (b) materialism. As predicted, both the GRAT and its subscales correlated positively with a measure of spiritual transcendence and negatively wit...