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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 29th 2015

    Looking at prime field, I notice that \mathbb{Q} is not listed as an example. Is that not usually considered a prime field?

    If the prime fields are to be considered a weakly initial set for the category of fields (as mentioned at field), then it needs to be included. I think we can also describe a prime field as a field admitting an epi from the initial object \mathbb{Z} in the category of commutative rings. Or, as a residue field of a localization of \mathbb{Z} with respect to a prime ideal (the case of \mathbb{Q} corresponding to the zero ideal being a prime ideal).

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeJun 29th 2015
    • (edited Jun 29th 2015)

    Lang [Algebra, 3rd ed.] writes:

    If KK is a field, then KK has characteristic 00 or p>0p \gt 0. In the first case, KK contains as a subfield an isomorphic image of the rational numbers, and in the second case, it contains an isomorphic image of 𝔽 p\mathbb{F}_p. In either case, this subfield will be called the prime field (contained in KK).

    So it appears to me that \mathbb{Q} is supposed to be a prime field.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 29th 2015

    I thought it was too.

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