Ineed Q 17 8. Derive the following results:a) The identity element of a subfield is the same as that of the field.b) If (F, +, ) is an indexed collection of subfields of the field (F,+, ), then(n ., +,) is also a subfield of (F, +, ).9. Let f be a homomorphism from the field (F, +, ) into itself and K be the set ofelements left fixed by f:K = {a E F|S(a)= a}.Given K (0}, verify that the triple (K,+,) is a subfield of (F,+, ).10. a) Consider the subset SCR' defined by S {a+ bVp|a, bEQ; pa prime}.Show that (S, +, ) is a subfield of (R',+, ).b) Prove that any subfield of (R', +, :) must contain the rational numbers.11. Prove that if the field (F,+,) is of characteristic p > 0, then every subfield of(F, +, ) has characteristic p.12. Let f be a homomorphism of the ring (R, +,) into the ring (R', +', ) and sup-pose (R, ,) has a subring (F,+,) which is a field.FC ker (f) or else (R',-+', ) contains a subring isomorphic to (F,+, ).Show that cither13. If R = {a + bv2|a, bE Z}, then the system (R,+, ) is an integral domain,but not a field. Obtain the field of quotients of (R,+, ).14. Suppose the integral domain (R,+,) is imbedded in the field (F',+'. '), say(R, +, ) (R',+', ') under the mapping f. Define the set K byK = {a' ' (b')-1 | a', b' E R'; &' 0).Prove (1) (K,+', ) is a subfield of (F, +', ) and (2) (K, +', ) is isomorphicto the field of quotients of (R, +,). [Hint: For (2), consider the function g definedby g(la, b]) = f(a) ' f(b)- where a, bE R, b 0.]15. Show that any field is isomorphic to its field of quotients. [Hint: Make use ofthe previous exercise with f as the identity map.]16. Prove that if (R,+, ) and (R', +', ') are isomorphic integral domains, then theirfields of quotients are also isomorphic.17. From Problem 8(h), deduce that every field (F, t,) has a unique prime subfield.Is this result still true if (F,+,) is assumed merely to be a division ring?

Name: Ineed Q 17 8. Derive the following results:a) The identity element of
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Description: Ineed Q 17 8. Derive the following results:a) The identity element of a subfield is the same as that of the field.b) If (F, +, ) is an indexed collection of subfields of the field (F,+, ), then(n ., +,) is also a subfield of (F, +, ).9. Let f be a ho