College Algebra Exam Review 334

# College Algebra Exam Review 334 - 344 7 FIELD EXTENSIONS...

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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. pp (a) Q. p 7/ 2; (b) Q.i; 7/ p 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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