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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 29th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexLooking 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).

   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).

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeJun 29th 2015
   • (edited Jun 29th 2015)
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   Author: Zhen Lin
   Format: MarkdownItexLang [_Algebra_, 3rd ed.] writes: > If $K$ is a field, then $K$ has characteristic $0$ or $p \gt 0$. In the first case, $K$ contains as a subfield an isomorphic image of the rational numbers, and in the second case, it contains an isomorphic image of $\mathbb{F}_p$. In either case, this subfield will be called the **prime field** (contained in $K$). So it appears to me that $\mathbb{Q}$ is supposed to be a prime field.

   Lang [Algebra, 3rd ed.] writes:

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

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

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 29th 2015
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   Author: Mike Shulman
   Format: MarkdownItexI thought it was too.

   I thought it was too.