Question

11. a) Prove that every field is a principal ideal domain. b) Show that the ring R nontrivial ideal of R. fa +bf2a, b e Z) is not a field by exhibiting a 12. Let fbe a homomorphism from the ring R into the ring R' and suppose that R ker for else R' contains has a subring F which is a field. Establish that either F a subring isomorphic to F 13. Derive the following results: a) The identity element of a subfield is the same as that of the field. b) If (Fi) is an index collection of subfields of the field F, then n Fi is also a subfield of F c) A subring F of a field F is a subfield of F if and only if F contains at least one d) A subset F of a finite field F is a subfield of F if and only if F contains more nonzero element and a e F for every nonzero a e F. than one element and is closed under addition and multiplication. 14, a) Consider the subset S of R# defined by s-la + b/pla, b e Q: p a fixed prime) Show that S is a subield of R b) Prove that any subhield of the field R" must contain the rational numbers 15. Prove that if the field F is of characteristic p>0, then every subfield of F has

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Answer 1

M is a proper ideal of a ring R. (i) suppose M is a maximal ideal of R let r NotElement M. Then rR is an ideal of R and hence M + rR is an ideal of R containing M properly. So, MCM + rR CR. Since M is maximal and M notequalto M + rR (since r NotElement M) M + rR = R Since I elementof R. These exists m elementof m and x element R such that m + r lambda = 1 implies 1 = m + r lambda implies 1 - r lambda = m implies 1 + r (- lambda) = m implies 1 + r a = m, where omega = -lambda elementof R implies 1 + r a = m elementof m (since m elementof m) Hence 1 + ra element m. Thus if m is a maximal ideal of R. than for every r NotElement M, there exists same a elementof R such that 1 + ra elementof M. (ii) Conversely suppose, for every r NotElement M, these exists