College Algebra Exam Review 334

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online