The simple-field conjecture via strongly IAC and VAC fields

Let KK be a field whose theory is simple. A field is strongly IAC if every field elementarily equivalent to it in the language of rings is IAC, and VAC means that it is dense in its algebraic closure with respect to every valuation.

Simple-field reduction conjecture.

  1. KK is strongly IAC; and
  2. If KK is VAC, then KK is pseudo-algebraically closed.

Since pseudo-algebraically closed fields and separably closed fields are VAC, this is proposed as a more manageable decomposition of the simple-field conjecture; the source gives no resolution status.

Sources & referencesView supporting material

Primary source

Peter Sinclair, “Immediately algebraically closed fields”, arXiv:1902.05830 (2020).

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