The Normal Basis Theorem — Galois Theory (Weintraub)
📐 Today’s Lesson: The Normal Basis Theorem From: Galois Theory (Weintraub) — Section 15
The Normal Basis Theorem
The Primitive Element Theorem tells us that a Galois extension E/F of degree n has the form E = F(α), so the powers 1, α, …, αⁿ⁻¹ form a basis. But there is an even more natural basis: one obtained by applying all n elements of the Galois group to a single element. The Normal Basis Theorem, proved in Section 3.6 of Weintraub, guarantees that such a basis always exists.
📖 Definition — Normal Basis (Definition 3.6.1)
A basis {αᵢ}ᵢ₌₁ⁿ of E over F is a normal basis if there exists α ∈ E such that {αᵢ} = {σᵢ(α)}, where {σᵢ} = Gal(E/F).
In other words, a normal basis is a basis consisting of all Galois conjugates of a single element. Note that being a normal basis generator is a stronger condition than being a primitive element: for a primitive element, the conjugates must be distinct; for a normal basis, they must be linearly independent.
✏️ Example — Remark 3.6.8: Primitive does not imply normal
Consider E = ℚ(√(2)) over F = ℚ. Let α = √(2). The conjugates are √(2) and -√(2), which are distinct (so α is primitive) but are not linearly independent over ℚ.
However, α = 1 + √(2) has conjugates 1 + √(2) and 1 - √(2), which are linearly independent, giving a normal basis.
The Cyclic Case
📐 Theorem — Theorem 3.6.2: Normal basis for cyclic extensions
Let σ generate Gal(E/F) and let T: E → E be given by T(α) = σ(α). Then Tⁿ - I = 0.
Claim: The minimum polynomial m_T(X) = Xⁿ - 1. If not, we would have a nontrivial linear relation among id, σ, …, σᵈ⁻¹ for d < n, contradicting the linear independence of distinct characters (Theorem 2.8.4).
Since m_T = n, there exists α ∈ E whose annihilator ideal is generated by m_T. Then {α, σ(α), …, σⁿ⁻¹(α)} are linearly independent and form a normal basis.
If E/F is a finite Galois extension with cyclic Galois group, then E has a normal basis.
The General Case
The proof for arbitrary (non-cyclic) Galois groups uses deeper linear algebra. A key ingredient is the following matrix criterion.
📐 Theorem — Lemma 3.6.3: Matrix criterion for bases
If {αⱼ} is a basis and a linear combination of rows vanishes, then Σ cᵢ σᵢ = 0 on all of E, contradicting the independence of distinct characters. Conversely, if not a basis, there is a nontrivial linear relation making columns dependent.
Let {σᵢ} = Gal(E/F). Then {α₁, …, αₙ} is a basis for E/F iff the matrix A = (σᵢ(αⱼ)) is nonsingular.
For infinite base fields, Weintraub proves a powerful algebraic independence result (Theorem 3.6.5): the elements of the Galois group are algebraically independent as functions on E. This is used in the proof of the general theorem.
📐 Theorem — Theorem 3.6.7: The Normal Basis Theorem
Finite fields: The extension is cyclic (Theorem 3.3.1), so Theorem 3.6.2 applies directly.
Infinite fields: Let Gal(E/F) = {σ₁, …, σₙ}. Construct the matrix B with entry bᵢⱼ = Xₖ where σᵢ σⱼ = σₖ (the group determinant). Computing (B) at (1, 0, …, 0) gives ± 1, so (B) is a nonzero polynomial in the indeterminates.
By algebraic independence of the σᵢ (Theorem 3.6.5), there exists α ∈ E with (B)(σ₁(α), …, σₙ(α)) ≠ 0. The resulting matrix A = (σᵢ(σⱼ(α))) is nonsingular, so {σ₁(α), …, σₙ(α)} is a basis by Lemma 3.6.3.
Let E be a finite Galois extension of F. Then E has a normal basis over F.
✏️ Example — Normal bases over finite fields
For 𝔽ₚ_ⁿ/𝔽ₚ with Galois group generated by Frobenius Φ(a) = aᵖ, a normal basis has the form:
String.raw{ α, αᵖ, αᵖ^², …, αᵖ^ⁿ^⁻^¹ }
Such a basis is particularly useful in applications to coding theory and cryptography, since squaring (or p-th power) in this basis amounts to a cyclic shift of coordinates.
The Normal Basis Theorem can also be stated as saying that E is a free module of rank 1 over the group algebra F[G], where G = Gal(E/F). A normal basis element α is a generator of this module. This perspective connects Galois theory to representation theory.
Key Takeaways
• 1.
A normal basis for E/F consists of all Galois conjugates of a single element. It exists for every finite Galois extension. • 2.
Being a normal basis generator is stronger than being a primitive element: the conjugates must be linearly independent, not just distinct. • 3.
For cyclic extensions, the proof uses the linear independence of characters. For the general case, it uses the group determinant and algebraic independence. • 4.
The matrix (σᵢ(αⱼ)) is nonsingular iff {αⱼ} is a basis – this is the key criterion connecting Galois theory to linear algebra. • 5.
Over finite fields, normal bases enable efficient arithmetic since the Frobenius acts as a cyclic coordinate shift.
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