Transcription of GALOIS THEORY AT WORK: CONCRETE EXAMPLES
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GALOIS THEORY AT WORK: CONCRETE EXAMPLES
GALOIS THEORY AT work : CONCRETE EXAMPLESKEITH field extensionQ( 2, 3)/Qis GALOIS of degree 4, so its GALOIS grouphas order 4. The elements of the GALOIS group are determined by their values on 2 and 3. TheQ-conjugates of 2 and 3 are 2 and 3, so we get at most four possibleautomorphisms in the GALOIS group. See Table 1. Since the GALOIS group has order 4, these4 possible assignments of values to ( 2) and ( 3) all really exist. ( 2) ( 3) 2 3 2 3 2 3 2 3 Table nonidentity automorphism in Table 1 has order 2. Since Gal(Q( 2, 3)/Q) con-tains 3 elements of order 2,Q( 2, 3) has 3 subfieldsKisuch that [Q( 2, 3) :Ki] = 2,or equivalently [Ki:Q] = 4/2 = 2. Two such fields areQ( 2) andQ( 3). A third isQ( 6) and that completes the list. Here is a diagram of all the ( 2, 3)Q( 2)Q( 3)Q( 6)QIn Table 1, the subgroup fixingQ( 2) is the first and second row, the subgroup fixingQ( 3) is the first and third row, and the subgroup fixingQ( 6) is the first and fourth row(since ( 2)( 3) = 2 3).
GALOIS THEORY AT WORK: CONCRETE EXAMPLES 3 Remark 1.3. While Galois theory provides the most systematic method to nd intermedi-ate elds, it may be possible to argue in other ways. For example, suppose Q ˆFˆQ(4 p 2) with [F: Q] = 2. Then 4 p 2 has degree 2 over F. Since 4 p 2 is a root of X4 2, its minimal polynomial over Fhas to be a ...
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