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Diff — Algebraic number field

Revision #2514 → #3169 · back to history

addedIdeal class group9226ee39e4a6
addedNorm of a prime idealbd5fccf6ef08
addedDedekind zeta generalizes Riemann zetab2b283f2d347
addedDirichlet L-functionb444405902d1
modifiedTrace and norm computationa2c3bdaf5caf
FieldFrom #2514To #3169
anchors[{"section":"Example","snippet":"We proceed to calculate the trace"},{"type":"math_alttext","value":"{\\displaystyle \\{1,\\zeta _{3}{\\sqrt[{3}]{2}},(\\zeta _{3}{\\sqrt[{3}]{2}})^{2}\\}}"},{"type":"math_alttext","value":"{\\displaystyle x=a+b\\zeta _{3}{\\sqrt[{3}]{2}}+c(\\zeta _{3}{\\sqrt[{3}]{2}})^{2}=a+b\\theta +c\\theta ^{2}.}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}xy=a(y_{0}+y_{1}\\theta +y_{2}\\theta ^{2})+\\\\b(2y_{2}+y_{0}\\theta +y_{1}\\theta ^{2})+\\\\c(2y_{1}+2y_{2}\\theta +y_{0}\\theta ^{2}).\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{bmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\\\a_{31}&a_{32}&a_{33}\\end{bmatrix}}{\\begin{bmatrix}y_{0}\\\\y_{1}\\\\y_{2}\\end{bmatrix}}={\\begin{bmatrix}ay_{0}+2cy_{1}+2by_{2}\\\\by_{0}+ay_{1}+2cy_{2}\\\\cy_{0}+by_{1}+ay_{2}\\end{bmatrix}}}"},{"type":"math_alttext","value":"{\\displaystyle A(x)={\\begin{bmatrix}a&2c&2b\\\\b&a&2c\\\\c&b&a\\end{bmatrix}}}"}][{"section":"Example","snippet":"We proceed to calculate the trace"},{"type":"math_alttext","value":"{\\displaystyle \\{1,\\zeta _{3}{\\sqrt[{3}]{2}},(\\zeta _{3}{\\sqrt[{3}]{2}})^{2}\\}}"},{"type":"math_alttext","value":"{\\displaystyle x=a+b\\zeta _{3}{\\sqrt[{3}]{2}}+c(\\zeta _{3}{\\sqrt[{3}]{2}})^{2}=a+b\\theta +c\\theta ^{2}.}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}xy=a(y_{0}+y_{1}\\theta +y_{2}\\theta ^{2})+\\\\b(2y_{2}+y_{0}\\theta +y_{1}\\theta ^{2})+\\\\c(2y_{1}+2y_{2}\\theta +y_{0}\\theta ^{2}).\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{bmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\\\a_{31}&a_{32}&a_{33}\\end{bmatrix}}{\\begin{bmatrix}y_{0}\\\\y_{1}\\\\y_{2}\\end{bmatrix}}={\\begin{bmatrix}ay_{0}+2cy_{1}+2by_{2}\\\\by_{0}+ay_{1}+2cy_{2}\\\\cy_{0}+by_{1}+ay_{2}\\end{bmatrix}}"},{"type":"math_alttext","value":"{\\displaystyle A(x)={\\begin{bmatrix}a&2c&2b\\\\b&a&2c\\\\c&b&a\\end{bmatrix}}}"}]
addedAbsolute value on a fieldea3a4ab21105
addedCompletion at a place525169b048cb
addedHensel's lemma21d8c3031e5f
addedInertia groupc2925651d1aa
addedGalois cohomologya1154862305d
addedDirichlet's unit theorem7e638f4efcbb
addedMinkowski's theoreme4bd51398321