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❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ▲❛❥♦s ❘ó♥②❛✐ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦ ❆❆◆❚✱ ❘■❈❆▼✱ ▲✐♥③✱ ◆♦✈❡♠❜❡r ✷✽✱ ✷✵✶✸ ▲❛❥♦s ❘ó♥②❛✐ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦

❈♦♥t❡♥ts ❇❛❝❦❣r♦✉♥❞ ♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ❛❧❣❡❜r❛s ❖r❞❡rs ❙♣❧✐tt✐♥❣ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r Q ❊①t❡♥s✐♦♥ t♦ ♥✉♠❜❡r ✜❡❧❞s ❆♣♣❧✐❝❛t✐♦♥s ❙♦♠❡ ✐♠♣r♦✈❡♠❡♥ts ❖♣❡♥ ♣r♦❜❧❡♠s ▲❛❥♦s ❘ó♥②❛✐ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦

❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ♠♦❞❡❧ ❊①❛❝t ✭s②♠❜♦❧✐❝✮ ❝♦♠♣✉t❛t✐♦♥s ♦❢ s✉❜str✉❝t✉r❡s ■♥♣✉t ❛♥❞ r❡s✉❧ts ✭❛❧❣❡❜r❛s✮ ✇✐❧❧ ❜❡ ❣✐✈❡♥ ❜② ❜❛s❡s ●r♦✉♥❞ ✜❡❧❞s✿ ❆❧❣❡❜r❛✐❝ ♥✉♠❜❡r ✜❡❧❞s✿ K = Q [ ① ] / ( ❢ ) ✱ ❢ ( ① ) ∈ Z [ ① ] ✐rr❡❞✉❝✐❜❧❡ ❋✐♥✐t❡ ✜❡❧❞s✿ K = F q ✳ ●❧♦❜❛❧ ❢✉♥❝t✐♦♥ ✜❡❧❞s✿ K = F q ( ① ) ✱ ❛♥❞ ✜♥✐t❡ ❡①t❡♥s✐♦♥s ❙✐③❡ ❂ ♥✉♠❜❡r ♦❢ ❜✐ts r❡♣r❡s❡♥t✐♥❣ t❤❡ ♦❜❥❡❝t ❲❡ ❝♦♥s✐❞❡r ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠s ✭❞❡t❡r♠✐♥✐st✐❝✱ r❛♥❞♦♠✐③❡❞✱ ✛✮ ▲❛❥♦s ❘ó♥②❛✐ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦

❙tr✉❝t✉r❡ ❝♦♥st❛♥ts r❡♣r❡s❡♥t❛t✐♦♥ ▲❡t A ❜❡ ❛♥ ❛❧❣❡❜r❛ ♦✈❡r K ✇✐t❤ ❜❛s✐s ❛ ✶ , . . . , ❛ ♠ ✳ ❚❤❡♥ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❝❛♥ ❜❡ s♣❡❝✐✜❡❞ ❜② ❣✐✈✐♥❣ ❛ ✐ · ❛ ❥ = γ ✐❥ ✶ ❛ ✶ + γ ✐❥ ✷ ❛ ✷ + · · · + γ ✐❥♠ ❛ ♠ . ❚❤❡ γ ✐❥❦ ∈ K ❛r❡ t❤❡ str✉❝t✉r❡ ❝♦♥st❛♥ts ✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ t❛❜❧❡ ♦❢ t❤❡ ❜❛s✐s✳ ❊ss❡♥t✐❛❧❧② t❤❡ r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ✭♣♦ss✐❜❧② ✈✐❛ ❉♦rr♦❤✬s ❡①t❡♥s✐♦♥✮✿ ♦♥❡ ♠❛② ❛ss✉♠❡ A ≤ ▼ ♠ + ✶ ( K ) . ▲❛❥♦s ❘ó♥②❛✐ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦

❈♦♠♣✉t❛t✐♦♥❛❧ t♦♦❧s ❛r✐t❤♠❡t✐❝ ♦✈❡r K ❧✐♥❡❛r ❛❧❣❡❜r❛ ✭s♦❧✈✐♥❣ s②st❡♠s ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✮ ♦✈❡r K ❢❛❝t♦r✐♥❣ ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r K ✭❇❡r❧❡❦❛♠♣✱ ▲▲▲✱ ✈❛♥ ❍♦❡✐❥✮ ❛♣♣r♦①✐♠❛t❡ ❝♦♠♣✉t❛t✐♦♥s ✇✐t❤ r❡❛❧ ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡rs ❛♣♣r♦①✐♠❛t❡ ❧❛tt✐❝❡ ❜❛s✐s r❡❞✉❝t✐♦♥ ✭▲▲▲✱ ❇✉❝❤♠❛♥♥✮ ▲❛❥♦s ❘ó♥②❛✐ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦

❈♦♠♣✉t✐♥❣ t❤❡ r❛❞✐❝❛❧ ♦❢ A ≤ ▼ ♥ ( K ) ❆♥ ❛ ∈ A ✐s ♥✐❧♣♦t❡♥t✱ ✐❢ ❛ ❞ = ✵ ❢♦r s♦♠❡ ❞ ✳ ❘❛❞ ( A ) ✐s t❤❡ ❧❛r❣❡st ✐❞❡❛❧ ♦❢ A ❝♦♥s✐st✐♥❣ ♦❢ ♥✐❧♣♦t❡♥t ❡❧❡♠❡♥ts✳ P♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠s ❜❛s❡❞ ♦♥ ❧✐♥❡❛r ❛❧❣❡❜r❛ ❚❤❡♦r❡♠ ✭❉✐❝❦s♦♥✬✷✸✮ ❆ss✉♠❡ ❝❤❛r K = ✵ ✳ ❚❤❡♥ A ✐s ♥✐❧♣♦t❡♥t ⇔ ∀ ❛ ∈ A , ❚r ( ❛ ) = ✵ . ❘❛❞ ( A ) = { ❛ ∈ A| ∀ ❜ ∈ A ∪ { ■ } , ❚r ( ❜❛ ) = ✵ } . ❈♦r♦❧❧❛r② ❆ss✉♠❡ ❝❤❛r K = ✵ ✱ ❇ = ( ❛ ❜❛s✐s ♦❢ A ) ∪ { ■ } ✳ ❚❤❡♥ ❘❛❞ ( A ) = { ❛ ∈ A| ∀ ❜ ∈ ❇ , ❚r ( ❜❛ ) = ✵ } . ▲❛❥♦s ❘ó♥②❛✐ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦

❈♦♠♣✉t✐♥❣ t❤❡ r❛❞✐❝❛❧ ♦❢ A ≤ ▼ ♥ ( K ) ❖✈❡r K = F ♣ ❬❋r✐❡❞❧✱ ❘ ✬✽✺❪✿ ❈♦♠♣✉t❡ ❛ s❡q✉❡♥❝❡ ♦❢ ✐❞❡❛❧s A = A ✵ ⊇ A ✶ ⊇ A ✷ ⊇ . . . ⊇ A ⌈ ❧♦❣ ♣ ♥ ⌉ = ❘❛❞ ( A ) ❊①t❡♥s✐♦♥s K = F q ❬❊❜❡r❧② ✬✽✾❪ K = F q ( ① ) ❬■✈❛♥②♦s✱ ❘✱ ❙③á♥tó ✬✾✹❪ ❆r❜✐tr❛r② K ❬❈♦❤❡♥✱ ■✈❛♥②♦s✱ ❲❛❧❡s ✬✾✻❪ → A ✐ + ✶ : s②st❡♠ ♦❢ s❡♠✐❧✐♥❡❛r ❡q✉❛t✐♦♥s ♦✈❡r K A ✐ − ❊q✉❛t✐♦♥s✿ ❢r♦♠ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❝❤❛r✳ ♣♦❧②♥♦♠✐❛❧ ❘❛❞✐❝❛❧ ✭♥✐❧✱ s♦❧✈❛❜❧❡✮ ❛❧❣♦r✐t❤♠s ❢♦r ▲✐❡ ❛❧❣❡❜r❛s ❬❘✬✾✵❪ ▲❛❥♦s ❘ó♥②❛✐ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦

❲❡❞❞❡r❜✉r♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ❆ss✉♠❡ ❘❛❞ ( A ) = ( ✵ ) ✳ ❚❤❡♥ A ✐s t❤❡ ❞✐r❡❝t s✉♠ ♦❢ ✐ts ♠✐♥✐♠❛❧ ✐❞❡❛❧s✿ A = A ✶ ⊕ A ✷ ⊕ · · · ⊕ A ❦ . ❚❤✐s ✐s r❡✢❡❝t❡❞ ✐♥ t❤❡ ❝❡♥tr❛❧ ♣❛rt✿ ❩ ( A ) = ❩ ( A ✶ ) ⊕ ❩ ( A ✷ ) ⊕ · · · ⊕ ❩ ( A ❦ ) . ▲❡t ✵ � = ❡ ✐ ∈ ❩ ( A ✐ ) ✳ ❚❤❡♥ A ✐ = ❡ ✐ A . ❋✐♥❞✐♥❣ t❤❡ ❩ ( A ✐ ) ✿ r❡❞✉❝❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ t♦ ❢❛❝t♦r✐♥❣ ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r K ▲❛❥♦s ❘ó♥②❛✐ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦

❆❧❣♦r✐t❤♠s ❢♦r ❲❡❞❞❡r❜✉r♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ❖✈❡r ✜♥✐t❡ ✜❡❧❞s ❛♥❞ ♥✉♠❜❡r ✜❡❧❞s✿ ❬❋r✐❡❞❧✱ ❘ ✬✽✺❪ ✖ ❛♥ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠ ❖✈❡r s✉✣❝✐❡♥t❧② ❧❛r❣❡ ✜❡❧❞s ❬❊❜❡r❧② ❵✾✶❪ ✖ ♦♥❡ r♦✉♥❞✱ r❛♥❞♦♠✐③❡❞ P✐❝❦ ❛ r❛♥❞♦♠ ❛ ∈ ❩ ( A ) ✱ ❧❡t ❢ ❜❡ t❤❡ ♠✐♥✳ ♣♦❧✳ ♦❢ ❛ ✳ ❚❤❡♥ ❩ ( A ) ∼ = K [ ① ] / ( ❢ ( ① )) ❢❛❝t♦rs ♦❢ ❢ ↔ ❲❡❞❞❡r❜✉r♥ ❝♦♠♣♦♥❡♥ts ♦❢ A ❖✈❡r ❣❧♦❜❛❧ ❢✉♥❝t✐♦♥ ✜❡❧❞s✿ ❬■✈❛♥②♦s✱ ❘✱ ❙③á♥tó ✬✾✹❪ r❡❞✉❝❡❞ t♦ ❢❛❝t♦r✐♥❣ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r t❤❡ ♣r✐♠❡ ✜❡❧❞ ▲❛❥♦s ❘ó♥②❛✐ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦