Mathematics 4321                       Review Test #1                             Spring 2007

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:

(a)   The square of any element in D is positive.

(b)   The identity is positive.

(c)    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.