9.4. MORE ON ALGEBRAIC FIELD EXTENSIONS 227

The following picture is a summary of what has just been shown.

Lk ≡K=KG(K,F) G(K,F) ≃ G(K,F)/G(K,K)...

......

L j =KG(L j ,F) G(L j,F) ≃ G(K,F)/G(K,L j)

......

...L1 =KG(L1,F) G(L1,F) ≃ G(K,F)/G(K,L1)

F≡ L0 G(L0,F) = {ι} ≃ G(K,F)/G(K,F)

9.4.6 PermutationsAs explained above, the automorphisms of a splitting fieldK of p(x)∈ F [x] are determinedby the permutations of the roots of p(x) . Thus it makes sense to consider permutations.

Let {a1, · · · ,an} be a set of distinct elements. Then a permutation of these elements isusually thought of as a list in a particular order. Thus there are exactly n! permutations ofa set having n distinct elements. With this definition, here is a simple lemma.

Lemma 9.4.33 Every permutation can be obtained from every other permutation by a fi-nite number of switches.

Proof: This is obvious if n = 1 or 2. Suppose then that it is true for sets of n− 1 ele-ments. Take two permutations of {a1, · · · ,an} ,P1,P2. To get from P1 to P2 using switches,first make a switch to obtain the last element in the list coinciding with the last element ofP2. By induction, there are switches which will arrange the first n−1 to the right order. ■

It is customary to consider permutations in terms of the set In ≡ {1, · · · ,n} to be morespecific. Then one can think of a given permutation as a mapping σ from this set In to itselfwhich is one to one and onto. In fact, σ (i)≡ j where j is in the ith position. Often peoplewrite such a σ in the following form(

1 2 · · · ni1 i2 · · · in

)(9.26)

meaning 1→ i1,2→ i2, ... where {i1, i2, ..., in} = {1,2, ...,n}. An easy way to understandthe above permutation is through the use of matrix multiplication by permutation matrices.The above vector (i1, · · · , in)T is obtained by

(ei1 ei2 · · · ein

)

12...n

 (9.27)

This can be seen right away from looking at a simple example or by using the definition ofmatrix multiplication directly.

Definition 9.4.34 The sign of the permutation 9.26 is defined as the determinant of theabove matrix in 9.27.