Math 721

Spring 2001

Homework 5

 

  1. Let be a field and an algebraic element of odd degree over  Show that

 

  1. Let be an extension of fields with intermediate extensions.

a)      Show that  is finite if and only if both  are finite.

b)      Show that if  is finite then both  divide

c)      Show that

d)      Show that if   are finite and relatively prime, then equality holds in part c.

e)      Show that if are algebraic over  then so is

 

  1. Compute the Galois group of  over , where  is the smallest field containing  and all the roots of

 

  1. Let be an extension of fields and let  be an intermediate ring with identity (that is, ).

a)      Show that  is an integral domain.

b)      Show that the following two conditions are equivalent:

·        Every intermediate domain  is a field.

·         is an algebraic extension of

 

5.      Construct all fields of order 4, 8, and 12 up to isomorphism (if they exist). Prove that your list is complete.