Let L / K L/K L / K field extension
of degree n = [ L : K ] n=[L:K] n = [ L : K ] α ∈ L \alpha\in L α ∈ L K K K m α : L → L m_\alpha:L\to L m α : L → L x ↦ α x x\mapsto \alpha x x ↦ αx (field) norm of α \alpha α L L L K K K
N L / K ( α ) : = det ( m α ) ∈ K .
\mathrm{N}_{L/K}(\alpha) := \det(m_\alpha)\in K.
N L / K ( α ) := det ( m α ) ∈ K . If L / K L/K L / K separable extension
) and Ω \Omega Ω L L L
N L / K ( α ) = ∏ σ σ ( α ) ,
\mathrm{N}_{L/K}(\alpha)=\prod_{\sigma} \sigma(\alpha),
N L / K ( α ) = σ ∏ σ ( α ) , where the product runs over all K K K embeddings
σ : L ↪ Ω \sigma:L\hookrightarrow \Omega σ : L ↪ Ω N L / K ( α β ) = N L / K ( α ) N L / K ( β ) \mathrm{N}_{L/K}(\alpha\beta)=\mathrm{N}_{L/K}(\alpha)\mathrm{N}_{L/K}(\beta) N L / K ( α β ) = N L / K ( α ) N L / K ( β ) trace/norm in towers
for a tower of fields
K ⊆ M ⊆ L K\subseteq M\subseteq L K ⊆ M ⊆ L
Examples Quadratic extension. Let L = K ( d ) L=K(\sqrt{d}) L = K ( d ) c h a r ( K ) ≠ 2 \mathrm{char}(K)\neq 2 char ( K ) = 2 α = a + b d \alpha=a+b\sqrt{d} α = a + b d
N L / K ( α ) = ( a + b d ) ( a − b d ) = a 2 − b 2 d .
\mathrm{N}_{L/K}(\alpha)=(a+b\sqrt{d})(a-b\sqrt{d})=a^2-b^2d.
N L / K ( α ) = ( a + b d ) ( a − b d ) = a 2 − b 2 d . Norm via minimal polynomial. If L = K ( α ) L=K(\alpha) L = K ( α ) simple extension
and the minimal polynomial of α \alpha α K K K m α ( x ) = x n + c n − 1 x n − 1 + ⋯ + c 0 m_\alpha(x)=x^n+c_{n-1}x^{n-1}+\cdots+c_0 m α ( x ) = x n + c n − 1 x n − 1 + ⋯ + c 0
N L / K ( α ) = ( − 1 ) n c 0 .
\mathrm{N}_{L/K}(\alpha)=(-1)^n c_0.
N L / K ( α ) = ( − 1 ) n c 0 . (Here c 0 c_0 c 0
Finite fields. For L = F q n L=\mathbb{F}_{q^n} L = F q n K = F q K=\mathbb{F}_q K = F q
N L / K ( α ) = α ⋅ α q ⋅ α q 2 ⋯ α q n − 1 = α ( q n − 1 ) / ( q − 1 ) .
\mathrm{N}_{L/K}(\alpha)=\alpha\cdot \alpha^{q}\cdot \alpha^{q^2}\cdots \alpha^{q^{n-1}}=\alpha^{(q^n-1)/(q-1)}.
N L / K ( α ) = α ⋅ α q ⋅ α q 2 ⋯ α q n − 1 = α ( q n − 1 ) / ( q − 1 ) . This is compatible with the cyclic structure of the multiplicative group (see finite-field multiplicative group is cyclic
).