Corollary 3.2.2. We have
Φ(Ω) = Λ.
Proof. Clearly Φ(Ω) ⊆ Λ. Further, dimQ Φ(Ω) = d, and by Lemma 3.2.1,
dimQ Λ = rank Q Λ = rank L Λ ≤ d.
Hence Φ(Ω) = Λ.
80

Definition 3.2.3. A subspace W of Ld is called special if σ̂ −1 (σ(x)) ∈ W for all x ∈
W, σ ∈ G.
Lemma 3.2.4. Let W be an L-linear subspace of Ld . Then W is special ⇐⇒ W is
generated by W ∩ Λ. Further in this case, we have dimL W = dimQ W ∩ Λ.
Proof. For the first implication, let a1 , . . . , ar be a Q-basis of W ∩ Λ. Let x ∈ W and
write x =

�r

i=1 λi ai

σ̂

−1

with λi ∈ L for i = 1, . . . , r. Then

(σ(x)) =

r
�

σ(λi )σ̂

−1

(σ(ai )) =

i=1

r
�
i=1

σ(λi )ai ∈ W for σ ∈ G.
So W is special. Further, in that case we have by Lemma 3.2.1
dimL W = rank L W ∩ Λ = dimQ W ∩ Λ.
For the second implication, assume W is special. Let [L : Q] = m and take a Q-basis
{w1 , . . . , wm } of L. Write G = {σ1 , . . . , σm }. Let x ∈ W and put
yi =

m
�

σj (wi )σ̂j−1 (σj (x)) for i = 1, . . . , m.

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This text contains a mathematical definition of what makes a subspace "special". This could be queried directly.

### Query ###
What is the mathematical definition of a special subspace?
What is the mathematical definition of a special subspace?