Note that it uses HTML tags for subscripts and superscripts, so it is difficult to decipher with an earlier viewer. Each definition is worth 5 points. Note the choice in questions 7 and 8.
G is normal, then multiplication of left cosets is well-defined.
Show by a direct computation that N is a normal subgroup of D 4 . Prove that any finite extension field is an algebraic extension. Let K be a subfield of E , and let E be a subfield of F . Prove that if E is an algebraic extension of K and F is an algebraic extension of E , then F is an algebraic extension of K . B. Prove that any nonzero prime ideal of a principal ideal domain is maximal.