Linear Algebra, Theory And Applications, 2012a

468 FIELDS AND FIELD EXTENSIONS
This is because σ (1) = 1 if σ is an automorphism. The original system in (6.18) is of the
form
σ 1 (u 1 ) x 1 + ···+ σ 1 (u k )1+···+ σ 1 (u l ) x l + ···+ σ 1 (u m ) x m =0
σ 2 (u 1 ) x 1 + ···+ σ 2 (u k )1+···+ σ 1 (u l ) x l + ···+ σ 2 (u m ) x m =0
.
σ n (u 1 ) x 1 + ···+ σ n (u k )1+···+ σ 1 (u l ) x l + ···+ σ n (u m ) x m =0
Now replace the k th equation with the difference of the k th equations in the original system
and the one in which σ r was done to both sides of the equations. Since σ r (x l ) ≠ x l the
result will be a linear system of the form My = 0 where y ≠ 0 has fewer nonzero entries
than x, contradicting the choice of x.
With the above estimate, here is another relation between the fixed fields and subgroups
of automorphisms. It doesn’t seem to depend on anything being a splitting field of a
separable polynomial.
Proposition F.5.11 Let H be a finite group of automorphisms defined on a field K. Then
for K H the fixed field,
G (K, K H )=H
Proof: If σ ∈ H, then by definition, σ ∈ G (K, K H ). It is clear that H ⊆ G (K, K H ) .
Then by Proposition F.5.10 and Theorem F.5.2,
|H| ≥[K : K H ] ≥|G (K, K H )|≥|H|
and so H = G (K, K H ).
This leads to the following interesting correspondence in the case where K is a splitting
field of a separable polynomial over a field F.
Fixed fields
L β → G (K, L)
K H
α
← H
Subgroups of G (K, F) (6.19)
Then αβL = L and βαH = H. Thus there exists a one to one correspondence between the
fixed fields and the subgroups of G (K, F). The following theorem summarizes the above
result.
Theorem F.5.12 Let K be a splitting field of a separable polynomial over a field F. Then
there exists a one to one correspondence between the fixed fields K H for H asubgroupof
G (K, F) and the intermediate fields as described in the above. H 1 ⊆ H 2 if and only if
K H1 ⊇ K H2 .Also
|H| =[K : K H ]
Proof: The one to one correspondence is established above. The claim about the fixed
fields is obvious because if the group is larger, then the fixed field must get harder because it
is more difficult to fix everything using more automorphisms than with fewer automorphisms.
Consider the estimate. From Theorem F.5.10, |H| ≥[K : K H ]. But also, H = G (K, K H )
from Proposition F.5.11 G (K, K H )=H and from Theorem F.5.2,
|H| = |G (K, K H )|≤[K : K H ] .
Note that from the above discussion, when K is a splitting field of p (x) ∈ F [x] , this
implies that if L is an intermediate field, then it is also a fixed field of a subgroup of G (K, F).
In fact, from the above,
L = K G(K,L)