Dickson algebras are atomic at $p$

TL;DR

This paper investigates the atomicity property of Dickson algebras within algebraic topology, revealing that not all invariant rings possess this property and linking it to specific Steenrod operations.

Contribution

It demonstrates that atomicity in Dickson algebras depends on the existence of Steenrod operations transforming monomials into p-th powers of generators.

Findings

Atomicity varies among invariant rings.

Existence of specific Steenrod operations determines atomicity.

Not all rings of invariants are atomic.

Abstract

The notion of atomicity defined by Cohen, Moore and Neisendorfer is studied for the Dickson algebras. Not any ring of invariants respects this property. It depends on the property of the Dickson algebra that given any monomial $d$ there exists a sequence of Steenrod operations $P(\Gamma, d) $ such that $P(\Gamma, d) d$ becomes a $p$-th power of the top Dickson algebra generator.