11. (Exercise 15.3) Let F be a field and p(x) an irreducible polynomial in F[r]. In this investigation we showed that E = Fr]/(p(z)) is a field, and we implied that F is a subfield of E. Now we will examine what we mean by that statement. (a) There is a natural mapping from F to E. Identify this mapping ( is called the inclusion mapping). Show that preserves the structure of F. Is an isomorphism? Explain. (b) Explain how E contains an isomorphic copy of F. (It is in this sense that we say F is a subfield of E. This subfield of E that is isomorphic to F is called an embedding of F in E.)

11. (Exercise 15.3) Let F be a field and p(x) an irreducible polynomial in F[r]. In this investigation we showed that E = Fr]/(p(z)) is a field, and we implied that F is a subfield of E. Now we will examine what we mean by that statement. (a) There is a natural mapping from F to E. Identify this mapping ( is called the inclusion mapping). Show that preserves the structure of F. Is an isomorphism? Explain. (b) Explain how E contains an isomorphic copy of F. (It is in this sense that we say F is a subfield of E. This subfield of E that is isomorphic to F is called an embedding of F in E.)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.6: Algebraic Extensions Of A Field

Problem 7E

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