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 F., +,) is also a subfield of (F,+,).
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 F., +,) is also a subfield of (F,+,).
Transcribed Image Text: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 of
elements 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 cither
13. 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 by
K = {a' ' (b')-1 | a', b' E R'; &' 0).
Prove (1) (K,+', ) is a subfield of (F, +', ) and (2) (K, +', ) is isomorphic
to the field of quotients of (R, +,). [Hint: For (2), consider the function g defined
by 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 of
the previous exercise with f as the identity map.]
16. Prove that if (R,+, ) and (R', +', ') are isomorphic integral domains, then their
fields 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?
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![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 of
elements 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 cither
13. 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 by
K = {a' ' (b')-1 | a', b' E R'; &' 0).
Prove (1) (K,+', ) is a subfield of (F, +', ) and (2) (K, +', ) is isomorphic
to the field of quotients of (R, +,). [Hint: For (2), consider the function g defined
by 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 of
the previous exercise with f as the identity map.]
16. Prove that if (R,+, ) and (R', +', ') are isomorphic integral domains, then their
fields 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?](https://content.bartleby.com/qna-images/question/2230d3d2-01e3-465a-b1bd-d58628b96aab/918c309b-3d00-4af0-a6cb-788485b4a82e/4pnjug_processed.jpeg)
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