College Algebra Exam Review 312

College Algebra Exam Review 312 - 322 7 FIELD EXTENSIONS...

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322 7. FIELD EXTENSIONS – FIRST LOOK 7.3. Adjoining Algebraic Elements to a Field The fields in this section are general, not necessarily subfields of the com- plex numbers. A field extension L of a field K is, in particular, a vector space over K . You are asked to check this in the Exercises. It will be helpful to review the definition of a vector space in Section 3.3 Since a field extension L of a field K is a K -vector space, in particular, L has a dimension over K , possibly infinite. The dimension of L as a K - vector space is denoted dim K .L/ (or, sometimes, OEL W .) Dimensions of field extensions have the following multiplicative property: Proposition 7.3.1. If K L M are fields, then dim K .M/ D dim K .L/ dim L .M/: Proof. Suppose that f 1 ; : : : ; r g is a subset of L that is linearly inde- pendent over K , and that f 1 ; : : : ; s g is a subset of M that is linearly independent over L . I claim that f i j W 1 i r; 1 j s g is linearly independent over K . In fact, if 0 D P i;j k ij i j D P j . P i k ij i / j , with k ij 2 K , then linear independence of f j g over L implies that P i k ij i D 0 for all j , and then linear independence of f i g over K implies that k ij D 0 for all i; j , which proves the claim.
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