Math 421/621

Spring 2000

Homework 5

 

  1. Let be the real cube root of 2 and let be a root of the polynomial . Show that the fields QQ. (Note that these fields are not the same, the left one is real, but the right is not contained in the real numbers).

 

  1. Let F and K be fields with K a finite extension of F. Let D be an integral domain such that

a)      Show that K is an algebraic extension of F.

b)      Show that D is a field.

c)      Give an examplewhere F and K are arbitrary fields and D is an integral domain that is not a field.

 

  1. Consider the field F=Q

a)      Show that [F:Q]=6

b)      Exhibit subfields of F of degrees 1,2,3 and 6 over Q.

c)      Are there subfields of F of degree 4 or 5 over Q?

 

  1. Let  be an odd number. Show that

 

  1. (G) If are algebraic over K of degrees m and n respectively, then show:

a)  

b)   If  gcd(m,n)=1 then