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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeNov 26th 2017

    I added to field a mention of some other constructive variants of the definition, with a couple more references.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 26th 2017

    I fixed up the DOI link for Johnstone’s paper (at field and at local ring), since the geniuses at Elsevier use parentheses in their DOIs, which break markdown syntax.

    For reference, one needs to replace (nn) in the DOI url by %28nn%29.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeNov 26th 2017

    Thanks! I’ve noticed that before, but always forget about it.

  1. Clarify a sentence to say that the discrete field condition implies that 010\neq 1.

    Egbert Rijke

    diff, v49, current

    • CommentRowNumber5.
    • CommentAuthorUlrik
    • CommentTimeJan 8th 2021

    Change definition of denial field to actually match MRR87

    diff, v50, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2021

    cross-linked the discussion of weakly initial sets in the category of fields with the corresponding example at multi-adjoint

    diff, v53, current

  2. one pair of authors defined fields to be possibly trivial.

    Anonymous

    diff, v57, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2022
    • (edited May 26th 2022)

    I have taken the liberty of turning this comment from a sub-section into a remark (here). Also tried to adjust the wording a little, for readability.

    diff, v58, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2022
    • CommentRowNumber10.
    • CommentAuthorMarkSaving
    • CommentTimeJul 3rd 2022

    There appears to be an error in the section on construction notions of a field. Specifically, the claim is that a residue field is discrete iff equality is decidable. But this seems not to be true.

    In fact, the statement “A residue field with decidable equality is a Heyting field” is equivalent to excluded middle.

    To see this, consider a proposition pp. Consider the set R p={xp}R_p = \mathbb{Z} \cup \{x \in \mathbb{Q} \mid p\}, which is a subring of \mathbb{Q}. Since R pR_p is a subset of \mathbb{Q} and \mathbb{Q} has decidable equality, R pR_p also has decidable equality. And of course 010 \neq 1 in R pR_p.

    I claim that R pR_p is a residue field iff ¬¬p\neg \neg p. For suppose ¬¬p\neg \neg p, and consider some xR px \in R_p. Suppose xx does not have a multiplicative inverse. Now suppose x0x \neq 0. Then we see that x 1R px^{-1} \notin R_p. If pp held, we would have x 1R px^{-1} \in R_p. So we know ¬p\neg p holds. But this is a contradiction. Therefore, xx must be zero (using decidable equality).

    Conversely, suppose RR is a residue field. Then 202 \neq 0, so 2 does not fail to have an inverse. That is, 2 12^{-1} is not not in R pR_p. Then ¬¬p\neg \neg p.

    I claim that R pR_p is a Heyting field iff pp iff R pR_p is a discrete field. For suppose RR is a Heyting field. Then either 2 or 3 has a multiplicative inverse, so either 2 12^{-1} or 3 1R p3^{-1} \in R_p. In either case, we see that pp holds. If pp holds, then R p=R_p = \mathbb{Q}, which is a discrete field. And if R pR_p is a discrete field, it is clearly a Heyting field.

    With these facts in hand, we see that if every residue field with decidable equality is a Heyting field, then p¬¬pp \iff \neg \neg p holds for all pp. So we have full excluded middle.

    Of course, assuming excluded middle, it is clear that all residue fields with decidable equality are discrete fields. \square

    I have not yet determined whether all Heyting fields with decidable equality are discrete, but it seems very likely that this also cannot be proven.

    If no one objects, I will change the page to say that a residue field is discrete if and only if unithood is decidable.

  3. added Mark Saying’s proof that not every residue field with decidable equality is Heyting.

    Anonymous

    diff, v61, current

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