9.1. FINITE AND ALGEBRAIC EXTENSIONS421finite field extension, by Proposition7.3.9. Proposition7.3.1implies thatKK.a0; : : : ; an; b/is a finite field extension.It then follows fromProposition7.3.4thatK.a0; : : : ; an; b/is algebraic overK, so, in particu-lar,bis algebraic overK.Part (b) is a consequence of part (a).nDefinition 9.1.2.Suppose thatKLis a field extension and thatEandFare intermediate fields,KELandKFL. ThecompositeEFofEandFis the smallest subfield ofLcontainingEandF.FEFL[j[jKEProposition 9.1.3.Suppose thatKLis a field extension and thatEandFare intermediate fields,KEL, andKFL.(a)IfEis algebraic overKandFis arbitrary, thenEFis alge-braic overF.(b)IfEandFare both algebraic overK, thenEFis algebraicoverK.(c)dimF.EF /dimK.E/.Proof.Exercises
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