A Counterexample to Hilbert's Fourteenth Problem in Dimension 5

A Linear Counterexample to the Fourteenth Problem of Hilbert in Dimension Eleven

A family of Ga-actions on affine space Am is constructed, each having a non-finitely generated ring of invariants (m ≥ 6). Because these actions are of small degree, they induce linear actions of unipotent groups Ga Ga on A 2n+3 for n ≥ 4, and these invariant rings are also non-finitely generated. The smallest such action presented here is for the group Ga Ga acting linearly on A11.

A Resolution to Hilberts First Problem

The continuum hypothesis (CH) is one of and if not the most important open problems in set theory, one that is important for both mathematical and philosophical reasons. The general problem is determining whether there is an infinite set of real numbers that cannot be put into one-to-one correspondence with the natural numbers or be put into one-to-one correspondence with the real numbers respe...

Counterexample to Hilbert's fourteenth problem for the 3-dimensional additive group

Consistency of a counterexample to Naimark's problem.

We construct a C*-algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any Hilbert space. This answers an old question of Naimark. Our construction uses a combinatorial statement called the diamond principle, which is known to be consistent with but not provable from the standard axioms of set theory (assumin...

Counterexamples to the Fourteenth Problem of Hilbert

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...