Please post carefully solutions of the following ones:
1. Let be a field extension of . By defining scalar multiplication for and by , the multiplication in , show that is an vector space.
2. If is a field extention of , prove that iff .
3. Let be a field extension of , and let . Show that the evaluation map given by is a ring and an vector space homomorphism. (Such a map is called an algebra homomorphism.)
4. Let be a field extension of and let . Then and ,
so is the quotient field of .
5. Show that .
6. Verify the following universal mapping property for polynomial rings:
(a) Let be a ring containing a field . If , show that there is a unique ring homomorphism with for each .
(b) Moreover, suppose that is a ring containing , together with a function , satisfying the following property: For any ring containing and elements , there is a unique ring homomorphism with . Show that is isomorphic to .
7. Let be a ring. If is also an vector space and forall and , then is said to be an algebra. If is an algebra , show that contains an isomorphic copy of . Also show that if is a field extension of , then is an algebra.
8. Let be a finite extension of . For , let be the map from to defined by . Show that is an linear transformation. Also show that is the minimal polynomial of . For which is ?
9. If is an extension of such that is prime, show that there are no intermediate fields between and .
10. If is a field extension of and if such that is odd, show that . Given an example to show that this can be false if the degree of over is even.
11. If is an algebraic extension of and if is a subring of with , show that is a field.
12. Show that and are not isomorphic as fields but are isomorphic as vector spaces over .
13. If and , show that the composite is equal to .
14. If and are field extensions of that are cotained in a common field, show that is a finite extension of iff both and are finite extensions of .
15. If and are field extensions of that are cotained in a common field, show that is algebraic over iff both and are algebraic over .
16. Let be the algebraic closure of in . Prove that .
17. Let be a finite extension of . If and are subfields of containing , show that . If , prove that .
18. Show that .
19. Given an example of field extensions of for which .
20. Given an example of a field extension with but with for any .
21. Let be a root of , where . Show that factors as , where .
22. (a)Let be field , and let . If and , let . Prove that is irreducible over iff is irreducible over for any .
(b)Show that is irreducible over if is a prime.
(Hint: Replace by and use the Eisenstein criterion.)
23. Recall that the characteristic of a ring with identity is the smallest positive integer for which , if such an exists, or else the characteristis is . Let be a ring with identity. Define by , where is the identity of . Show that is a ring homomorphism and that for a unique nonnegative integer , and show that is the characteristis of .
24. For any positive integer , given an example of a ring of characteristis .
25. If is an integral domain, show that either or is prime.
26. Let be a commutative ring with identity. The prime subring of is the intersection of all subrings of . Show that this intersection is a subring of that is contained inside all subrings of . Moreover, show that the prime subring of is equal to , where is the multiplicative identity of .
27. Let be a field. If , show that the prime subring of is isomorphic to the field , and if , then the prime subring is isomorphic to .
28. Let be a field. The prime subfield of is the intersection of all subfileds of . Show that this subfield is the quotient field of the prime subring of , that it is contained inside all subfileds of , and that it is isomorphic to or depending on whether the characteristis of is or .
Those are all problems in Section 1 of GTM 167.
Here is version pdf