Further afield and further, a field: remarks on undecidability

Abstract: Given a field $K$, one can ask "what first-order sentences are true in $K$"? E.g. for $K = \mathbb{C}$, "$\exists x (x^2 = -1)$" is true, but for $K = \mathbb{Q}$ this is false. One major area of study at the intersection of logic and number theory is, given a field $K$ of number-theoretic interest, whether there is an algorithmic process which can decide the truth or falsity of a given first-order sentence in $K$. For $K = \mathbb{C}$, there exists such an algorithmic process; for $K = \mathbb{Q}$ there cannot (due to work of Gödel & Julia Robinson).

I will pose a related question: whether the logical consequences of a given sentence in a field may be decided algorithmically. Often the answer is no; so e.g. we cannot algorithmically detect general properties of fields $K$ with a Galois extension $L$ such that $\mathrm{Gal}(L/K) \cong S_5$, or e.g. general properties of characteristic $p$ fields that admit points on a given rationally parameterisable curve over $\mathbb{F}_p$. I will focus on those fields whose behaviour is tightly controlled by their absolute Galois group, and prove some precise limitations.

I will aim for this talk to be self-contained on the logic side of things!

number theory

Audience: researchers in the topic

London number theory seminar

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For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html