Prove that C is a field.

Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.

Consider the set ={[0],[2],[4],[6],[8]}10, with addition and multiplication as defined in 10. a. Is R an integral domain? If not, give a reason. b. Is R a field? If not, give a reason. [Type here][Type here]

Consider the set S={[0],[2],[4],[6],[8],[10],[12],[14],[16]}18, with addition and multiplication as defined in 18. a. Is S an integral domain? If not, give a reason. b. Is S a field? If not, give a reason. [Type here][Type here]

Prove that any ordered field must contain a subfield that is isomorphic to the field of rational numbers.

14. a. If is an ordered integral domain, prove that each element in the quotient field of can be written in the form with in . b. If with in , prove that if and only if in .

Construct a field having the following number of elements.

If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]

[Type here] True or False Label each of the following statements as either true or false. 2. Every field is an integral domain. [Type here]

True or False Label each of the following statements as either true or false. Every field is a division ring.