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Unformatted text preview: 344 7. FIELD EXTENSIONS – FIRST LOOK 7.5.6. Verify that the following ﬁeld extensions of Q are Galois. Find a
polynomial for which the ﬁeld is the splitting ﬁeld. Find the Galois group,
the lattice of subgroups, and the lattice of intermediate ﬁelds.
(a) Q. p 7/
(b) Q.i; 7/ p
(c) Q. 2 C 7/
(d) Q.e 2 i=7 /
(e) Q.e 2 i=6 /
7.5.7. The Galois group G and splitting ﬁeld E of f .x/ D x 8 2 can be
analyzed in a manner quite similar to Example 7.5.14. The Galois group G
has order 16 and has a (normal) cyclic subgroup of order 8; however, G is
not the dihedral group D8 . Describe the group by generators and relations
(two generators, one relation). Find the lattice of subgroups. G has a
normal subgroup isomorphic to D4 ; in fact the splitting ﬁeld E contains
the splitting ﬁeld of x 4 2. The subgroups of D4 already account for a
large number of the subgroups of G . Describe as explicitly as possible the
lattice of intermediate ﬁelds between Q and the splitting ﬁeld.
7.5.8. Suppose E is a Galois extension of a ﬁeld K (both ﬁelds assumed,
for now, to be contained in C ). Suppose that the Galois group AutK .E/
is abelian, and that an irreducible polynomial f .x/ 2 KŒx has one root
˛ 2 E . Show that K.˛/ is then the splitting ﬁeld of f . Show by examples
that this is not true if the Galois group is not abelian or if the polynomial
f is not irreducible. ...
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- Fall '08