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 Algebra Notes
04-02-2015, 02:02 PM
Post: #2
 elim Moderator Posts: 581 Joined: Feb 2010 Reputation: 0
1.2 Prime subfields

1.2.1 Definition Call $\displaystyle{\bigcap\{S\mid S\subset_{\small F} F\}}$ the prime subfield of field $F$

1.2.2 Remark A prime subfield $\text{psub}(F)$ is a field.

1.2.3 Examples $\mathbb{Q},\;\mathbb{Z}_p\;$($p$ is prime) are their own prime subfields.

1.2.4 Proposition For any field $F,\;\text{psub}(F)$ is isomorphic to either $\mathbb{Z}_p$
$\qquad$for some prime $p$(in which case we say $F$ has characteristic $p$)
$\qquad$or to $\mathbb{Q}$ ($F$ has characteristic $0$ then).
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 Messages In This Thread Algebra Notes - elim - 04-02-2015, 11:29 AM 1.2 Prime subfields - elim - 04-02-2015 02:02 PM 1.3 Fields of fractions - elim - 04-02-2015, 04:40 PM 1.4 Polynomial rings - elim - 04-02-2015, 05:11 PM 1.5 Polynomial rings over fields - elim - 04-03-2015, 04:40 PM RE: Algebra Notes - elim - 04-04-2015, 08:22 PM Notes - elim - 04-05-2015, 07:51 AM Notes - elim - 04-05-2015, 08:02 AM Notes - elim - 04-06-2015, 07:05 AM 2 Field Extensions - elim - 04-06-2015, 07:24 AM Notes - elim - 04-08-2015, 02:58 PM Notes - elim - 04-08-2015, 02:58 PM

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