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- Prove that any ordered field must contain a subfield that is isomorphic to the field of rational numbers.Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .)
- 27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.True or False Label each of the following statements as either true or false. The field of rational numbers is an extension of the integral domain of integers.Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]