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College Algebra Exam Review 411

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

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9.1. FINITE AND ALGEBRAIC EXTENSIONS 421 finite field extension, by Proposition 7.3.9 . Proposition 7.3.1 implies that K K.a 0 ; : : : ; a n ; b/ is a finite field extension. It then follows from Proposition 7.3.4 that K.a 0 ; : : : ; a n ; b/ is algebraic over K , so, in particu- lar, b is algebraic over K . Part (b) is a consequence of part (a). n 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 E F L [j [j K E 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 alge- braic over F . (b) If E and F are both algebraic over K , then E F is algebraic over K . (c) dim F .E F / dim K .E/ . Proof. Exercises
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