Spring 2007 – Mathematics 4321 – Review Test #1

— Dr. Cordero

I. Definitions and examples- Give the definition and an example of each of the following:

1. ring, subring
2. commutative ring
3. ring with identity (unity)
4. zero divisor
5. integral domain
6. field
7. subfield
8. division ring
9. ring isomorphism
10. characteristic of a ring
11. ordered integral domain
12. well-ordered integral domain
13. field of quotients of an integral domain
14. ordered field
15. upper bound; least upper bound
16. complete ordered field
17. extension field
18. algebraic elements; transcendental elements
19. algebraic extension
20. algebraically closed field  
21. algebraic closure     

II. Theorems

1. Let D be a commutative ring with identity. Prove that D is an integral domain if and only if the cancellation laws hold in D.
2. State and prove the theorem that gives necessary and sufficient conditions for a subset of a ring to be a subring.
3. Prove that every field is an integral domain. Is the converse true? Prove or give a counterexample.
4. Is every finite integral domain a field? Prove or give a counterexample.
5. State and prove the theorem that gives necessary and sufficient conditions for a subset of a field to be a subfield.
6. What ring properties are preserved under a ring isomorphism? State the theorem that gives those. Study the proof of the theorem.
7. Prove that if D is an integral domain, then either char D=0 or char D=p where p is prime.
8. Let  D be  an ordered integral domain with identity e.
   Prove the following:
   1. The square of any element in D is positive.
   2. The identity is positive.
   3. If D is well-ordered, then the identity is the least positive element.
9. Prove that the square root of 2 is not a rational number.
10.  State the Fundamental Theorem of Algebra.
11. Study the statement of the following theorems:  24.1, 24.2, 25.1,  27.2, 27.3, 28.1 and its corollaries,   28.2, 29.1,  31.2,  31.3, 32.2.