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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 Â 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 Â L is a ﬁeld extension and that E and
F are intermediate ﬁelds, K Â E Â L and K Â F Â L. The composite
E F of E and F is the smallest subﬁeld of L containing E and F . F
E ÂL Proposition 9.1.3. Suppose that K Â L is a ﬁeld extension and that E
and F are intermediate ﬁelds, K Â E Â L, and K Â F Â 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.
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.
- Fall '08