域扩张(群论笔记)
域扩张
标签（空格分隔）： field extension 域扩张
Q ( 2 3 ) \mathbb{Q(\sqrt[3]{2})} Q(32)不是正规扩张
Q ( 2 3 , ω ) \mathbb{Q(\sqrt[3]{2}, \omega)} Q(32,ω)是正规扩张
ω = − 1 + 3 i 2 \omega=\frac{-1+\sqrt{3}i}{2} ω=2−1+3i
相得
Q ( 2 , 3 ) / Q ( 2 ) = { 1 , 3 } \mathbb{Q}(\sqrt{2}, \sqrt{3})/ \mathbb{Q}(\sqrt{2}) = \{1,\sqrt{3}\} Q(2,3)/Q(2)={1,3}
Q ( 2 ) / Q = { 1 , 2 } \mathbb{Q}(\sqrt{2})/ \mathbb{Q} = \{1,\sqrt{2}\} Q(2)/Q={1,2}
Q ( 2 , 3 ) / Q = { 1 , 2 , 3 , 6 } \mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}= \{1, \sqrt{2}, \sqrt{3}, \sqrt{6}\} Q(2,3)/Q={1,2,3,6}
Q ( 2 , 3 3 ) / Q ( 2 ) = { 1 , 3 3 , 3 2 3 } \mathbb{Q}(\sqrt{2}, \sqrt[3]{3})/ \mathbb{Q}(\sqrt{2}) = \{1,\sqrt[3]{3},\sqrt[3]{3^2}\} Q(2,33)/Q(2)={1,33,332}
Q ( 2 ) / Q = { 1 , 2 } \mathbb{Q}(\sqrt{2})/ \mathbb{Q} = \{1,\sqrt{2}\} Q(2)/Q={1,2}
Q ( 2 , 3 3 ) / Q = { 1 , 3 3 , 3 2 3 , 2 , 2 3 3 , 2 3 2 3 } \mathbb{Q}(\sqrt{2}, \sqrt[3]{3})/ \mathbb{Q} = \{1,\sqrt[3]{3},\sqrt[3]{3^2}, \sqrt{2},\sqrt{2}\sqrt[3]{3},\sqrt{2}\sqrt[3]{3^2}\} Q(2,33)/Q={1,33,332,2,233,2332}
共轭元
正规扩张每个元素的共轭元都包含在其中，相应的Galous群上就是共轭作用，而共轭作用不动的子群就是正规子群
求 6 + 10 + 15 \sqrt{6}+\sqrt{10}+\sqrt{15} 6+10+15 极小多项式 ϕ \phi ϕ
σ = { σ 1 ( 2 , 3 , 5 ) , σ 2 ( 2 , 3 , − 5 ) , σ 3 ( 2 , − 3 , 5 ) , σ 4 ( 2 , − 3 , − 5 ) , σ 5 ( − 2 , 3 , 5 ) , σ 6 ( − 2 , 3 , − 5 ) , σ 7 ( − 2 , − 3 , 5 ) , σ 8 ( − 2 , − 3 , − 5 ) } \sigma=\{ \sigma_1(\sqrt{2},\sqrt{3},\sqrt{5}),\\ \sigma_2(\sqrt{2},\sqrt{3},-\sqrt{5}),\\ \sigma_3(\sqrt{2},-\sqrt{3},\sqrt{5}),\\ \sigma_4(\sqrt{2},-\sqrt{3},-\sqrt{5}),\\ \sigma_5(-\sqrt{2},\sqrt{3},\sqrt{5}),\\ \sigma_6(-\sqrt{2},\sqrt{3},-\sqrt{5}),\\ \sigma_7(-\sqrt{2},-\sqrt{3},\sqrt{5}),\\ \sigma_8(-\sqrt{2},-\sqrt{3},-\sqrt{5}) \} σ={σ1(2,3,5),σ2(2,3,−5),σ3(2,−3,5),σ4(2,−3,−5),σ5(−2,3,5),σ6(−2,3,−5),σ7(−2,−3,5),σ8(−2,−3,−5)}
θ = 6 + 10 + 15 \theta=\sqrt{6}+\sqrt{10}+\sqrt{15} θ=6+10+15
σ 1 θ = σ 8 θ = 6 + 10 + 15 \sigma_1\theta = \sigma_8\theta=\sqrt{6}+\sqrt{10}+\sqrt{15} σ1θ=σ8θ=6+10+15
σ 2 θ = σ 7 θ = 6 − 10 − 15 \sigma_2\theta = \sigma_7\theta=\sqrt{6}-\sqrt{10}-\sqrt{15} σ2θ=σ7θ=6−10−15
σ 3 θ = σ 6 θ = − 6 + 10 − 15 \sigma_3\theta = \sigma_6\theta=-\sqrt{6}+\sqrt{10}-\sqrt{15} σ3θ=σ6θ=−6+10−15
σ 4 θ = σ 5 θ = − 6 − 10 + 15 \sigma_4\theta = \sigma_5\theta=-\sqrt{6}-\sqrt{10}+\sqrt{15} σ4θ=σ5θ=−6−10+15
ϕ = ( x − σ 1 θ ) ( x − σ 2 θ ) ( x − σ 3 θ ) ( x − σ 4 θ ) = x 4 − 62 x 2 − 240 x − 239 \phi = (x-\sigma_1\theta)(x-\sigma_2\theta)(x-\sigma_3\theta)(x-\sigma_4\theta)\\ =x^4-62x^2-240x-239 ϕ=(x−σ1θ)(x−σ2θ)(x−σ3θ)(x−σ4θ)=x4−62x2−240x−239
G a l ( Q ( 2 , 3 , 5 ) / Q ) Gal(\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})/\mathbb{Q}) Gal(Q(2,3,5)/Q)