We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecomposable, that is, they remain indecomposable in any generic extension. However, any absolutely rigid family of groups has cardinality less than the partition cardinal κ(ω). Added December 2004: The proofs of Theorems 0.2 and 0.3 are not correct, and the clai...

Jullien’s indecomposability theorem states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis, and then, in the base system RCA0 plus Σ 1 1 induction, it was shown by Neeman to have strength strictly between weak Σ 1 choice and ∆...

We construct indecomposable and noncrossed product division algebras over function fields of smooth curves X over Zp. This is done by defining an index preserving morphism s : Br(K̂(X)) → Br(K(X)) which splits res : Br(K(X)) → Br(K̂(X)), where K̂(X) is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over K̂(X).

Let K denote a field. A polynomial f(x) ∈ K[x] is said to be decomposable over K if f(x)= g(h(x)) for some polynomials g(x) and h(x)∈K[x] with 1< deg(h) < deg(f ). Otherwise f(x) is called indecomposable. If f(x)= g(xm) with m> 1, then f(x) is said to be trivially decomposable. In this paper, we show that xd+ax+b is indecomposable and that if e denotes the largest proper divisor of d, then xd+a...