College Algebra Exam Review 411

College Algebra Exam Review 411 - 9.1. FINITE AND ALGEBRAIC...

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Unformatted text preview: 9.1. FINITE AND ALGEBRAIC EXTENSIONS 421 finite field extension, by Proposition 7.3.9. Proposition 7.3.1 implies that K  K.a0 ; : : : ; an ; b/ is a finite field extension. It then follows from Proposition 7.3.4 that K.a0 ; : : : ; an ; b/ is algebraic over K , so, in particular, b is algebraic over K . Part (b) is a consequence of part (a). I Definition 9.1.2. Suppose that K  L is a field extension and that E and F are intermediate fields, K  E  L and K  F  L. The composite E F of E and F is the smallest subfield of L containing E and F . F [j K   EF [j E ÂL Proposition 9.1.3. Suppose that K  L is a field extension and that E and F are intermediate fields, K  E  L, and K  F  L. (a) If E is algebraic over K and F is arbitrary, then E F is algebraic over F . (b) If E and F are both algebraic over K , then E F is algebraic over K . (c) dimF .E F / Ä dimK .E/. Proof. Exercises 9.1.1 through 9.1.3. I Exercises 9.1 9.1.1. Prove Proposition 9.1.3 (a). Hint: Let a 2 E F . Then there exist ˛1 ; : : : ; ˛n 2 E such that a 2 F .˛1 ; : : : ; ˛n /. 9.1.2. Prove Proposition 9.1.3 (b). 9.1.3. Prove Proposition 9.1.3 (c). Hint: In case dimK .E/ is infinite, there is nothing to be done. So assume the dimension is finite and let ˛1 ; : : : ; ˛n be a basis of E over K . Conclude successively that E F D F .˛1 ; : : : ; ˛n /, then that E F D F Œ˛1 ; : : : ; ˛n , and finally that E F D spanF f˛1 ; : : : ; ˛n g. p p 9.1.4. What is the dimension of Q. 2 C 3/ over Q? ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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