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(iii) If R is a subﬁeld of k , prove that R is the prime ﬁeld of k .
Solution. Absent.
3.27 (i) Show that every subﬁeld of C contains Q.
Solution. Every subﬁeld of C contains 1, hence contains Z, and
hence contains Q, for it must contain the multiplicative inverse of
every nonzero integer. (ii) Show that the prime ﬁeld of R is Q.
Solution. This follows from (i).
(iii) Show that the prime ﬁeld of C is Q.
Solution. This follows from (i).
3.28 (i) For any ﬁeld F , prove that (2, F ) ∼ Aff(1, F ), where (2, F )
=
denotes the stochastic group.
Solution. The only properties of R used in setting up the isomorphism in Example 2.48(iv) is that it is a ﬁeld. (ii) If F is a ﬁnite ﬁeld with q elements, prove that
 (2, F ) = q (q − 1).
Solution. Now Aff(1, F ) consists of all functions f : x → ax + b,
where a , b ∈ F and a = 0, there are q − 1 choices for a and q
choices for b, and so  Aff(1, F ) = q (q − 1). But isomorphic
groups have the same order, and so  (2, F ) = q (q − 1).
(iii) Prove that (2, F3 ) ∼ S3 .
=
Solution. By part (ii),  (2, F3 ) = 6. By Proposition 2.135, it is
isomorphic to S3 or I6 ; as it is nonabelian, (2, F3 ) ∼ S3 .
=
3.29 True or false with reasons.
(i) The sequence notation for x 3 − 2x + 5 is (5, −2, 0, 1, 0, · · · ).
Solution. True.
(ii) If R is a domain, then R [x ] is a domain.
Solution. True.
(iii) Q [x ] is a ﬁeld.
Solution. False.
(iv) If k is a ﬁeld, then the prime ﬁeld of k [x ] is k .
Solution. True.
(v) If R is a domain and f (x ), g (x ) ∈ R [x ] are nonzero, then deg( f g ) =
deg( f ) + deg(g ).
Solution. True. ...
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 Fall '11
 KeithCornell
 Category theory, Isomorphism, Bijection, Group isomorphism

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