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Unformatted text preview: 342 7. FIELD EXTENSIONS – FIRST LOOK As the size of the Galois group is also 8, each of these assignments must
determine an element of the Galois group.
In particular, we single out two Q–automorphisms of E :
p
p
4
4
W 2 7! i 2; i 7! i;
and p
p
4
4
W 2 7! 2; i 7! i :
The automorphism is complex conjugation restricted to E .
Evidently, is of order 4, and is of order 2. Furthermore, we can
1 . It follows that the Galois group is generated
compute that
D
by and and is isomorphic to the dihedral group D4 . You are asked to
check this in the Exercises.
We identify p Galois group D4 as a subgroup of S4 , acting on the
the
r 1 4 2, 1 Ä r Ä 4. With this identiﬁcation,
roots ˛r D i
D .1234/ and
D .24/.
D4 has 10 subgroups (including D4 and fe g). The lattice of subgroups
is show in Figure 7.5.2; all of the inclusions in this diagram are of index 2.
D4
d d V0 h i D h.24/i
r Z4 D h i d
d h.13/i V d rr
r d
d
d
d
d
d d
d h.13/.24/i d
rr d
rd
rr
d
d
r fe g h.12/.34/i d
d h.14/.23/i ¨ ¨¨ ¨ ¨¨ ¨¨ ¨
¨ Figure 7.5.2. Lattice of subgroups of D4 . Here V denotes the group
V D fe; .12/.34/; .13/.24/; .14/.23/g;
and V 0 the group
V 0 D fe; .24/; .13/; .13/.24/ D 2 g: By the Galois correspondence, there are ten intermediate ﬁelds between Q and E , including the the ground ﬁeld and the splitting ﬁeld. Each ...
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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
 EVERAGE
 Algebra

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