Let F/K be a field extension and a, b E F be algebraic over K. If a has degree m over K and b 0 has degree n over K, prove that the elements a+b, ab, ab, ab-¹ have degree at most mn over K.
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Field Extension
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- 18. Let be the smallest subring of the field of rational numbers that contains . Find a description for a typical element of .Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].Corollary requires that be a field. Show that each of the following polynomials of positive degree has more than zeros over where is not a field. over over
- Since this section presents a method for constructing a field of quotients for an arbitrary integral domain D, we might ask what happens if D is already a field. As an example, consider the situation when D=5. a. With D=5, write out all the elements of S, sort these elements according to the relation , and then list all the distinct elements of Q. b. Exhibit an isomorphism from D to Q.If is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .Prove that if F is an ordered field with F+ as its set of positive elements, then F+nen+, where e denotes the multiplicative identity in F. (Hint: See Theorem 5.34 and its proof.) Theorem 5.34: Well-Ordered D+ If D is an ordered integral domain in which the set D+ of positive elements is well-ordered, then e is the least element of D+ and D+=nen+.
- Let be a field. Prove that if is a zero of then is a zero ofAssume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.Each of the polynomials in Exercises is irreducible over the given field . Find all zeros of in the field obtained by adjoining a zero of to . (In Exercises and , has three zeros in .)
- 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 .Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)In Exercises , a field , a polynomial over , and an element of the field obtained by adjoining a zero of to are given. In each case: Verify that is irreducible over . Write out a formula for the product of two arbitrary elements and of . Find the multiplicative inverse of the given element of . , ,