## Maps of Extensions and the Galois Group (Pt. II)

**Point of Post: **This is a continuation of this post.

The above illustrates a general principle concerning simple extensions:

**Theorem: ***Let be any field. Then, algebraic one has that is precisely the automorphisms sending to where is another root of in . In particular, is the number of roots of in .*

The fact that all of the described mappings are the only possible mappings is clear, and the fact that there really do exist automorphisms with the desired as image we merely define by mapping for any (this is well-defined because –then is an automorphism with the desired properties.

This automatically could have told us why is trivial.

An interesting corollary of the above is that for simple extensions one has that . This actually turns out to be true in more generality, but we need to develop an important lemma first.

Let be a monoid and a field. A *character *is a monoid homomorphism . Of course we have a -vector space of all mappings . Then, we claim that if is any distinct set of characters, they are necessarily linearly independent in . More precisely:

**Theorem(Dedekind’s Lemma): ***Let be characters. Then, if (as functions) then . *

**Proof: **Suppose not, then we may assume without loss of generality that no smaller subset of is linearly dependent. Now, there necessarily exists such that as functions. But, since there exists such that . But, we clearly have that

for all . Now, we must also have that

for all . Subtracting these two expressions gives

for all which contradicts that is linearly independent.

With this we can prove the fundamental result:

**Theorem: ***Let be finite. Then, .*

**Proof: **The idea is simple. Suppose that are the elements of (we clearly know that is finite, since is finite and so looks like and acts faithfully on the union of the roots of ). Suppose then that with basis , then necessarily there exists such that for all . This clearly contradicts Dedekind’s lemma with .

**References:**

[1] Morandi, Patrick. *Field and Galois Theory*. New York: Springer, 1996. Print.

[2] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[3] Lang, Serge. *Algebra*. New York: Springer, 2002. Print.

[4] Conrad, Keith. *Collected Notes on Field and Galois Theory*. Web. <http://www.math.uconn.edu/~kconrad/blurbs/>.

[5] Clark, Pete. *Field Theory*. Web. <http://math.uga.edu/~pete/FieldTheory.print

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