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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❛✐ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞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❛✐ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞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 = Fq✳

• ❧♦❜❛❧ ❢✉♥❝t✐♦♥ ✜❡❧❞s✿ K = Fq(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❛✐ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞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❛✐ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞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❛✐ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞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❛✐ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s

❈♦♠♣✉t✐♥❣ t❤❡ r❛❞✐❝❛❧ ♦❢ A ≤ ▼♥(K)

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

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

❙✐♠♣❧❡ ❛❧❣❡❜r❛s ♦✈❡r K

❚❤❡♦r❡♠ ✭❲❡❞❞❡r❜✉r♥✮ ❆ s✐♠♣❧❡ K✲❛❧❣❡❜r❛ A ✐s ✐s♦♠♦r♣❤✐❝ t♦ ▼❦(D)✱ ❢♦r ❛ ✭s❦❡✇✮✜❡❧❞ D ✇✐t❤ ❩(D) ≥ K ❛♥❞ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❦✳ ❊①♣❧✐❝✐t ✐s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ A✱ ✜♥❞ ❦✱ D ❛♥❞ ∼ =✳ K ✐s ✜♥✐t❡✿ ✕ ❬❘ ✬✾✵❪ r❛♥❞♦♠✐③❡❞ ♣♦❧② t✐♠❡ ✕ ❬❊❜❡r❧②✱ ●✐❡s❜r❡❝❤t ✬✾✼❪ r❛♥❞♦♠✐③❡❞ ≈ ❖(♥✸) ✕ ❬■✈❛♥②♦s✱ ❑❛r♣✐♥s❦✐✱ ❘✱ ❙❛①❡♥❛ ✬✶✶❪ ❞❡t❡r♠✐♥✐st✐❝ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❢♦r ❞✐♠ A ❜♦✉♥❞❡❞ K = R ♦r K = C✿ ✕ ❬❊❜❡r❧② ✬✾✶❪ r❛♥❞♦♠✐③❡❞ ♣♦❧② t✐♠❡ ✕ ❞❡r❛♥❞♦♠✐③❡❞ ❬❘❀ ❞❡ ●r❛❛❢✱ ■✈❛♥②♦s ✬✾✸✱ ✬✾✽❪

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

❊①♣❧✐❝✐t ✐s♦♠♦r♣❤✐s♠ ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s

❉✐✣❝✉❧t✿ ✭≻ ❢❛❝t♦r✐♥❣ ✐♥t❡❣❡rs✮ ❬❘ ✬✽✽❪✱ ❬❱♦✐❣❤t ✬✶✵❪ ❞✐♠ ❡A❡ ≺ ❢❛❝t♦r✐③❛t✐♦♥ ❬■✈❛♥②♦s✱ ❘ ✬✾✸❪ Pr♦❜❧❡♠✿ ✐s ❡①♣❧✐❝✐t ✐s♦♠♦r♣❤✐s♠ ≺ ❢❛❝t♦r✐③❛t✐♦♥❄ ✛✲❛❧❣♦r✐t❤♠✿ ❛❧❧♦✇❡❞ t♦ ❝❛❧❧ ❛♥ ♦r❛❝❧❡ ❢♦r ❢❛❝t♦r✐♥❣ ✐♥t❡❣❡rs✱ ❛♥❞ ❢♦r ❢❛❝t♦r✐♥❣ ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r ✜♥✐t❡ ✜❡❧❞s ❨❡s ❢♦r K = ◗✱ ❞✐♠ A = ✹ ❬■✈❛♥②♦s✱ ❙③á♥tó ✬✾✹❪ ❨❡s ❢♦r A = ▼♥(K)✱ ✇✐t❤ K ❛♥❞ ♥ s♠❛❧❧ ❬■✈❛♥②♦s✱ ❘✱ ❙❝❤✐❝❤♦ ✬✶✷❪ ■♠♣r♦✈❡♠❡♥t ❢♦r K = Q ❛♥❞ ♥ s♠❛❧❧ ❬▲❡❧❦❡s✱ ❘ ✬✶✸❪ ◆♦♥ ❝♦♥✈❡♥t✐♦♥❛❧ ❛♣♣❧✐❝❛t✐♦♥s✿ r❛t✐♦♥❛❧ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ✈❛r✐❡t✐❡s✱ ♥✲❞❡s❝❡♥t ❢♦r ❡❧❧✐♣t✐❝ ❝✉r✈❡s ❬❈r❡♠♦♥❛✱ ❋✐s❤❡r✱ ❖✬◆❡✐❧✱ ❙✐♠♦♥✱ ❙t♦❧❧✱ ✬✵✽✱ ✬✵✾✱ ✬✶✶❪

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

❖r❞❡rs ✐♥ s❡♠✐s✐♠♣❧❡ ❛❧❣❡❜r❛s ♦✈❡r K

■♥t❡❣r❛❧ ❡❧❡♠❡♥t✿ r♦♦t ✐♥ A ♦❢ ❛ ♠♦♥✐❝ ♣♦❧②♥♦♠✐❛❧ ✇✐t❤ ✐♥t❡❣❡r ❝♦❡✣❝✐❡♥ts ❖r❞❡r✿ ▲❛r❣❡ s✉❜r✐♥❣ ♦❢ A ❝♦♥s✐st✐♥❣ ♦❢ ✐♥t❡❣r❛❧ ❡❧❡♠❡♥ts ❋✉❧❧ ❧❛tt✐❝❡ ✐♥ A✿ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ Z✲s✉❜♠♦❞✉❧❡ ▼ ✇✐t❤ Q▼ = A ❖r❞❡r✿ ❛ s✉❜r✐♥❣ Λ ≤ A ✇✐t❤ ✶ ♦❢ A ✐♥ Λ s✉❝❤ t❤❛t Λ ✐s ❛ ❢✉❧❧ ❧❛tt✐❝❡ ✐♥ A ▼❛①✐♠❛❧ ♦r❞❡r✿ ♠❛①✐♠❛❧ ✇✳r✳t✳ ✐♥❝❧✉s✐♦♥ ✭♥♦t ✉♥✐q✉❡✮

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

❊①❛♠♣❧❡s ♦❢ ♦r❞❡rs

❚❤❡ s✉❜❧❛tt✐❝❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❜❛s✐s ❡❧❡♠❡♥ts ❛♥❞ ✶✱ ✐❢ t❤❡ str✉❝t✉r❡ ❝♦♥st❛♥ts ❛r❡ ✐♥t❡❣❡rs Z● ✐♥ t❤❡ ❣r♦✉♣ ❛❧❣❡❜r❛ Q● ❢♦r ❛ ✜♥✐t❡ ❣r♦✉♣ ● ❆❧❣❡❜r❛✐❝ ✐♥t❡❣❡r ♠❛tr✐❝❡s✿ K ♥✉♠❜❡r ✜❡❧❞✱ ❘ = {❛❧❣✳ ✐♥t❡❣❡rs ✐♥ K} ▼♥(❘) ✐♥ ▼♥(K) ✐s ❛ ♠❛①✐♠❛❧ ♦r❞❡r ❙t❛❜✐❧✐③❡rs ♦❢ ❧❛tt✐❝❡s✿ ▼ ❛ ❢✉❧❧ ❧❛tt✐❝❡ ✐♥ A O(▼) = {① ∈ A|①▼ ⊆ ▼}

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

❈♦♥str✉❝t✐♥❣ ♠❛①✐♠❛❧ ♦r❞❡rs

• ✐✈❡♥ ❛ s❡♠✐s✐♠♣❧❡ ❛❧❣❡❜r❛ A ♦✈❡r K✱ ✜♥❞ ❛ ♠❛①✐♠❛❧ ♦r❞❡r Λ ≤ A

❬■✈❛♥②♦s✱ ❘ ✬✾✸❪ ✕ ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✛✲❛❧❣♦r✐t❤♠ ❙t❛rts ✇✐t❤ ❛♥ ✐♥✐t✐❛❧ ♦r❞❡r Γ ❛♥❞ t❤❡♥ ❡♥❧❛r❣❡s ✐t ✐♥t♦ ❛ ♠❛①✐♠❛❧ ♦r❞❡r Λ ≥ Γ ❙✉♣♣♦s❡ t❤❛t A ∼ = ▼♥(D)✳ ❋r♦♠ Λ ♦♥❡ ❝❛♥ ❡✣❝✐❡♥t❧② ❞❡t❡r♠✐♥❡ ♥ ❯s❡ ♦❢ ❢❛❝t♦r✐♥❣✿ ✇❡ ♥❡❡❞ ♦♥❧② t❤❡ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ ❞(Γ)

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

• ♦❧❞❡♥ r✐♥❣s

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

• ♦❧❞❡♥ ❛❧❣❡❜r❛✱ ❣♦❧❞❡♥ r✐♥❣✱ ❣♦❧❞❡♥ ❝♦❞❡
• r♦✉♥❞ ✜❡❧❞✿ K = Q(✐) ✕ t❤❡ ●❛✉ss✐❛♥ r❛t✐♦♥❛❧s

GA = K(①, ✉; ①✷ = ✺, ✉✷ = ✐, ①✉ = −✉①) GO = Z[✐][①, ✉] ≤ GA t❤❡ ❣♦❧❞❡♥ r✐♥❣ ■t ❧❡❛❞s t♦ ❛ ❤✐❣❤❧② ❡✣❝✐❡♥t s♣❛❝❡✲t✐♠❡ r❛❞✐♦ ❝♦❞❡ ✭❨❛♦✱ ❲♦r♥❡❧❧❀ ❉❛②❛❧✱ ❱❛r❛♥❛s✐❀ ❇❡❧✜♦r❡✱ ❘❡❦❛②❛✱ ❱✐t❡r❜♦✮ GO ✐s ❛ ❧❛tt✐❝❡ ✇✐t❤ ✈❡r② s♠❛❧❧ ❞✐s❝r✐♠✐♥❛♥t ❝♦♥s✐st✐♥❣ ♦❢ ♥♦♥s✐♥❣✉❧❛r ✐♥t❡❣r❛❧ ♠❛tr✐❝❡s

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

• ♦❧❞❡♥ r✉s❤

❍♦❧❧❛♥t✐✱ ▲❛❤t♦♥❡♥✱ ❘❛♥t♦✱ ❱❡❤❦❛❧❛❤t✐✿ ✜♥❞ ♦r❞❡rs s✐♠✐❧❛r t♦ GO ✐♥ ♦t❤❡r ❝②❝❧✐❝ ❛❧❣❡❜r❛s ■❘✲❛❧❣♦r✐t❤♠✿ ❣✐✈❡♥ ❛♥ ♦r❞❡r Γ ≤ A ✐♥ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s✐♠♣❧❡ ❛❧❣❡❜r❛ A ♦✈❡r Q❀ ✐t ❝♦♠♣✉t❡s ❛ ♠❛①✐♠❛❧ ♦r❞❡r Λ ≥ Γ P♦❧②♥♦♠✐❛❧ t✐♠❡ ✛✲❛❧❣♦r✐t❤♠ ❞❡ ●r❛❛❢✿ ▼❆●▼❆ ✲✐♠♣❧❡♠❡♥t❛t✐♦♥ ❲✐t❤ t❤❛t ❜❡tt❡r ❝♦❞❡s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞✳ ❋♦r ❡①❛♠♣❧❡ ✐♥ GA+ = K(①, ✉; ①✷ = ✷ + ✐, ✉✷ = ✐, ①✉ = −✉①) t❤❡ ♦r❞❡r Γ = Z[✐][①, ✉] ✐s ♥♦t ♠❛①✐♠❛❧✳ ❚❤❡ ♠❛①✳ ♦r❞❡r Λ ⊃ Γ ♦❜t❛✐♥❡❞ ❜② t❤❡ ■❘✲❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛ ❜❡tt❡r ❝♦❞❡ ❊♥❣✐♥❡❡r✐♥❣ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ♥♦♥ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡r t❤❡♦r②

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

❚❤❡ ♠❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ▲❡t K ❜❡ ❛♥ ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡r ✜❡❧❞ ♦❢ ❞❡❣r❡❡ ❞ ❛♥❞ ❞✐s❝r✐♠✐♥❛♥t ∆ ♦✈❡r Q✳ ▲❡t A ❜❡ ❛♥ ❛ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛ ♦✈❡r K ❣✐✈❡♥ ❜② str✉❝t✉r❡ ❝♦♥st❛♥ts s✉❝❤ t❤❛t A ∼ = ▼♥(K) ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ♥✳ ❙✉♣♣♦s❡ t❤❛t ❞✱ ♥ ❛♥❞ |∆| ❛r❡ ❜♦✉♥❞❡❞✳ ❚❤❡♥ ❛♥ ✐s♦♠♦r♣❤✐s♠ A → ▼♥(K) ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✛✲❛❧❣♦r✐t❤♠✳ ❚❤❡ t✐♠❡ ❜♦✉♥❞ ♦❢ ♦✉r ❛❧❣♦r✐t❤♠ ❞❡♣❡♥❞s ♣♦❧②♥♦♠✐❛❧❧② ♦♥ |∆| ❛♥❞ ❡①♣♦♥❡♥t✐❛❧❧② ♦♥ ♥ ❛♥❞ ❞✳ Pr♦♦❢ ✇✐❧❧ ❜❡ ♦✉t❧✐♥❡❞ ❢♦r t❤❡ ❝❛s❡ K = Q ❚❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ❢♦❧❧♦✇s ❜② ❡①t❡♥❞✐♥❣ t❤❡ ❛r❣✉♠❡♥t ✈✐❛ t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣ K → R❞

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

❙♠❛❧❧ ③❡r♦ ❞✐✈✐s♦rs ✐♥ ♠❛①✐♠❛❧ ♦r❞❡rs

❚❤❡♦r❡♠ ▲❡t Λ ❜❡ ❛ ♠❛①✐♠❛❧ ♦r❞❡r ✐♥ A ≤ ▼♥(R)✱ A ∼ = ▼♥(Q)✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡❧❡♠❡♥t ❈ ∈ Λ ✇❤✐❝❤ ❤❛s r❛♥❦ ✶ ❛s ❛ ♠❛tr✐①✱ ❛♥❞ ✇❤♦s❡ ❋r♦❜❡♥✐✉s ♥♦r♠ ❈ ✐s ❧❡ss t❤❛♥ ♥✳ ❚❤❡ ❋r♦❜❡♥✐✉s ♥♦r♠ ♦❢ ❛ ♠❛tr✐① ❳ ∈ ▼♥(R) ✐s ❳ =

• ❚r(❳ ❚❳)

❲❡ ❤❛✈❡ Λ = P▼♥(Z)P−✶ ❢♦r s♦♠❡ P ∈ ●▲♥(R) ❉✐✈✐❞✐♥❣ ❜② | ❞❡t P|✶/♥ ✇❡ ♠❛② ❛ss✉♠❡ P ∈ ▼♥(R)✱ ❞❡t P = ±✶

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

Pr♦♦❢

▲❡t ρ ❜❡ t❤❡ ❧❡❢t ✐❞❡❛❧ ♦❢ ▼♥(Z) ♦❢ ❛❧❧ ♠❛tr✐❝❡s ✇❤✐❝❤ ❛r❡ ✵ ❡①❝❡♣t ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥ ρ ✐s ❛ ❧❛tt✐❝❡ ♦❢ ❝♦✈♦❧✉♠❡ ✶ ✐♥ t❤❡ s♣❛❝❡ ❙ ∼ = R♥ ♦❢ ❛❧❧ r❡❛❧ ♠❛tr✐❝❡s ❤❛✈✐♥❣ ❛❧❧ ③❡r♦s ♦✉ts✐❞❡ t❤❡ ✜rst ❝♦❧✉♠♥ ▲ = Pρ ✐s ❛ s✉❜❧❛tt✐❝❡ ♦❢ ❙✱ ✇✐t❤ ❝♦✈♦❧✉♠❡ ✶ ❆♣♣❧② ▼✐♥❦♦✇s❦✐✬s t❤❡♦r❡♠ t♦ ▲ ✐♥ ❙ ❛♥❞ t♦ t❤❡ ❜❛❧❧ ♦❢ r❛❞✐✉s √♥ ✐♥ ❙ ❝❡♥t❡r❡❞ ❛t t❤❡ ③❡r♦ ♠❛tr✐① ❚❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ❜❛❧❧ ✐s ♠♦r❡ t❤❛♥ ✷♥ ✕ ❝♦♥t❛✐♥s ✷♥ ✐♥t❡r♥❛❧❧② ❞✐s❥♦✐♥t ❝♦♣✐❡s ♦❢ t❤❡ ♥✲❞✐♠❡♥s✐♦♥❛❧ ✉♥✐t ❝✉❜❡ ❚❤❡r❡ ❡①✐sts ♥♦♥③❡r♦ ❇ ∈ ρ s✉❝❤ t❤❛t P❇ ❤❛s ❧❡♥❣t❤ < √♥

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

Pr♦♦❢✱ ❝♦♥❝❧✉❞❡❞

❇ ❛♥❞ ❤❡♥❝❡ P❇ ✐s ❛ r❛♥❦ ✶ ♠❛tr✐① ◆❡①t ❝♦♥s✐❞❡r t❤❡ tr❛♥s♣♦s❡ ♦❢ t❤✐s ❛r❣✉♠❡♥t ✇✐t❤ P−✶ ✐♥ t❤❡ ♣❧❛❝❡ ♦❢ P ❚❤❡r❡ ❡①✐sts ❛ ♥♦♥③❡r♦ ✐♥t❡❣❡r ♠❛tr✐① ❇′✱ ✇❤✐❝❤ ✐s ✵ ❡✈❡r②✇❤❡r❡ ❡①❝❡♣t ✐♥ t❤❡ ✜rst r♦✇✱ s✉❝❤ t❤❛t ❇′P−✶ ❤❛s ❧❡♥❣t❤ < √♥ ❈ := P❇❇′P−✶ ♠❡❡ts t❤❡ r❡q✉✐r❡♠❡♥ts ■t ✐s ✐♥ Λ ❜❡❝❛✉s❡ ❇❇′ ∈ ▼♥(Z)✱ ❛♥❞ ❈ = (P❇)(❇′P−✶) ≤ P❇ · ❇′P−✶ < (√♥)✷ = ♥ ❋✐♥❛❧❧② r❛♥❦ ❇❇′ = r❛♥❦ ❈ = ✶.

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

❍❡r♠✐t❡✬s ❝♦♥st❛♥t

γ♥ = s✉♣

▲

λ✶(▲)✷ (❞❡t ▲)✷/♥ , ✇❤❡r❡ ▲ ✐s ♦✈❡r ❛❧❧ ❢✉❧❧ ❧❛tt✐❝❡s ✐♥ R♥✱ ❛♥❞ λ✶(▲) ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❤♦rt❡st ♥♦♥③❡r♦ ✈❡❝t♦r ♦❢ ▲✳ √γ♥✿ ❞✐❛♠❡t❡r ♦❢ s♣❤❡r❡s ❛t t❤❡ ❞❡♥s❡st ❧❛tt✐❝❡ ❜❛s❡❞ ♣❛❝❦✐♥❣✳

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

♥ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✷✹ γ♥

♥

✶

✹ ✸

✷ ✹ ✽

✻✹ ✸

✻✹ ✷✽ ✹✷✹ ❋♦r ❧❛r❣❡ ♥ ✶ ✷π❡ ≤ γ♥ ♥ ≤ ✶.✼✹✺ ✷π❡ . ■♥ ❢❛❝t✱ ✇❡ ♣r♦✈❡❞ ❛❧s♦ ❈ ≤ γ♥✱ ❛♥❞ ❡✈❡♥ ❈ ≤ γ′

♥✱

✇❤❡r❡ γ′

♥ ✐s t❤❡ ❇❡r❣é✲▼❛rt✐♥❡t ❝♦♥st❛♥t✳

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

❚❤❡ ❇❡r❣é✲▼❛rt✐♥❡t ❝♦♥st❛♥t

γ♥ = s✉♣

▲

λ✶(▲)λ✶(▲∗) (❞❡t ▲)✷/♥ , ✇❤❡r❡ ▲ ✐s ♦✈❡r ❛❧❧ ❢✉❧❧ ❧❛tt✐❝❡s ✐♥ R♥✱ ❛♥❞ ▲∗ ✐s t❤❡ ❞✉❛❧ ❧❛tt✐❝❡ ♦❢ ▲ ▲∗ = {② ∈ R♥, ①, ② ∈ Z ❢♦r ❡✈❡r② ① ∈ ▲}. ❈❧❡❛r❧② γ′

♥ ≤ γ♥✳

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

❙♠❛❧❧ ✐s ❜❡❛✉t✐❢✉❧

▲❡♠♠❛ ✭❋✐s❤❡r ✬✵✼✮ ▲❡t ❳ ∈ ▼♥(C) ❜❡ ❛ ♠❛tr✐① s✉❝❤ t❤❛t ❞❡t ❳ ✐s ❛♥ ✐♥t❡❣❡r✱ ❛♥❞ ❳ < √♥✳ ❚❤❡♥ ❳ ✐s ❛ s✐♥❣✉❧❛r ♠❛tr✐①✳ Pr♦♦❢✳ ▲❡t ❳ = ◗❘ ❜❡ t❤❡ ◗❘ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❳✱ ◗ ✉♥✐t❛r②✱ ❘ ✉♣♣❡r tr✐❛♥❣✉❧❛r ✇✐t❤ ❞✐❛❣♦♥❛❧ ❡♥tr✐❡s r✶, r✷, . . . , r♥✳ ❲❡ ❤❛✈❡ | ❞❡t ❳|✷/♥ = (|r✶|✷|r✷|✷ · · · |r♥|✷)✶/♥ ≤ ≤ ✶ ♥(|r✶|✷ + |r✷|✷ + · · · |r♥|✷) ≤ ✶ ♥❘✷ = ✶ ♥❳✷ < ✶. ◆♦t❡ t❤❛t ❳ =

• ❚r(❳ ∗❳) =
• ❚r(❘∗❘) ❛s ◗∗◗ = ■✳ ❲❡

❝♦♥❝❧✉❞❡ t❤❛t ❞❡t ❳ = ✵✳

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

▲❡♠♠❛ ▲❡t ❳ ∈ ▼♥(Q) ❜❡ ❛ ♠❛tr✐① ✇❤♦s❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❤❛s ✐♥t❡❣r❛❧ ❝♦❡✣❝✐❡♥ts✱ ❛♥❞ ❳ < ✶✳ ❚❤❡♥ ❳ ✐s ❛ ♥✐❧♣♦t❡♥t ♠❛tr✐①✳ Pr♦♦❢✳ ❬●✳ ❑ós❪ ❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❳ ❛r❡ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡rs✱ ❤❡♥❝❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❳ t ❛r❡ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡rs ❛s ✇❡❧❧✱ ❢♦r ❛♥② t ∈ N+✳ ❲❡ ✐♥❢❡r t❤❛t ❳ t ❤❛s ❝❤❛r✳ ♣♦❧②✳ ✇✐t❤ ✐♥t❡❣r❛❧ ❝♦❡✣❝✐❡♥ts✳ ❚❤❡ ♥♦r♠ ❝♦♥❞✐t✐♦♥ ✐♠♣❧✐❡s ❳ t → ❖ ❛s t → ∞, ❤❡♥❝❡ ❳ t = ❖ ❢♦r t ❧❛r❣❡✳ ❚❤❡ ❜♦✉♥❞ ✐s s❤❛r♣✿ ❳ = ✶ ♥ ♥

✐,❥=✶

.

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

❘❡❞✉❝❡❞ ❜❛s❡s

Γ ✐s ❛ ❢✉❧❧ ❧❛tt✐❝❡ ✐♥ R♠ ✇✐t❤ ❜❛s✐s ❜✶, . . . , ❜♠ ♦✈❡r Z s✉❝❤ t❤❛t |❜✶| · |❜✷| · · · |❜♠| ≤ ❝♠ · ❞❡t(Γ) ❚❤❡ ▲▲▲ ❛❧❣♦r✐t❤♠ ❛❝❤✐❡✈❡s ❝♠ = ✷♠(♠−✶)/✹ ❚❤❡ ❛♣♣r♦①✐♠❛t❡ ✈❡rs✐♦♥ ♦❢ ▲▲▲ ❜② ❬❇✉❝❤♠❛♥♥ ✬✾✹❪ ❣✐✈❡s ❝♠ := (γ♠)

♠ ✷

✸ ✷ ♠ ✷

♠(♠−✶) ✷

γ♠ ✐s ❍❡r♠✐t❡✬s ❝♦♥st❛♥t

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

❙♠❛❧❧ ❝♦❡✣❝✐❡♥ts

▲❡♠♠❛ ✭❍✳ ❲✳ ▲❡♥str❛ ✬✽✸✮ Γ ✐s ❛ ❢✉❧❧ ❧❛tt✐❝❡ ✐♥ R♠ ✇✐t❤ ❜❛s✐s ❜✶, . . . , ❜♠ ♦✈❡r Z s✉❝❤ t❤❛t |❜✶| · |❜✷| · · · |❜♠| ≤ ❝♠ · ❞❡t(Γ) ❤♦❧❞s ❢♦r ❛ r❡❛❧ ♥✉♠❜❡r ❝♠ > ✵✳ ❙✉♣♣♦s❡ t❤❛t ✈ =

♠

• ✐=✶

α✐❜✐ ∈ Γ, α✐ ∈ Z. ❚❤❡♥ ✇❡ ❤❛✈❡ |α✐| ≤ ❝♠

|✈ | |❜✐| ❢♦r ✐ = ✶, . . . , ♠✳

❚❤❡ ♣r♦♦❢ ✐s ❈r❛♠❡r✬s r✉❧❡ ❛♥❞ ❍❛❞❛♠❛r❞✬s ❜♦✉♥❞✳

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

❚❤❡ ❛❧❣♦r✐t❤♠ ❢♦r K = Q

■♥♣✉t✿ A ∼ = ▼♥(Q)✱ ❣✐✈❡♥ ❜② ❛ ❜❛s✐s ✭♦❢ ♠ = ♥✷ ✈❡❝t♦rs✮ ❛♥❞ str✉❝t✉r❡ ❝♦♥st❛♥ts ♦✈❡r Q ❖✉t♣✉t✿ ❈ ∈ A✱ r❛♥❦ ❈ = ✶ ◆♦t❡ t❤❛t ▼ = A❈ ✐s ❛♥ ♥ ❞✐♠❡♥s✐♦♥❛❧ A ♠♦❞✉❧❡✱ ❤❡♥❝❡ ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥ ▼ ❣✐✈❡s ❛♥ A → ▼♥(Q) ✐s♦♠♦r♣❤✐s♠

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

❚❤❡ ❛❧❣♦r✐t❤♠ ❢♦r K = Q

✶ ❯s❡ t❤❡ ■❘ ❛❧❣♦r✐t❤♠ t♦ ❝♦♥str✉❝t ❛ ♠❛①✐♠❛❧ ♦r❞❡r Λ ✐♥ A✳ ✷ ❈♦♠♣✉t❡ ❛♥ A ֒

→ ▼♥(R)✳ ❯s❡ t❤❡ ❞❡r❛♥❞♦♠✐③❛t✐♦♥ ❢r♦♠ ❬❞❡

• r❛❛❢✱ ■✈❛♥②♦s ✬✵✵❪ ♦❢ ❬❊❜❡r❧② ✬✾✶❪✳ ❚❤✐s ❣✐✈❡s ❛ · ♦♥ A✳

✸ ❈♦♠♣✉t❡ ❛ s✉✣❝✐❡♥t❧② ♣r❡❝✐s❡ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❆ ♦❢ ♦✉r

❜❛s✐s ❇ ♦❢ Λ ❜② t❤❡ ♠❡t❤♦❞ ♦❢ ❬❙❝❤ö♥❤❛❣❡ ✬✽✷❪✳

✹ ❈♦♠♣✉t❡ ❛ r❡❞✉❝❡❞ ❜❛s✐s ❜✶, . . . , ❜♠ ♦❢ t❤❡ ❧❛tt✐❝❡ Λ ⊂ R♠ ❜②

t❤❡ ▲▲▲ ❛❧❣♦r✐t❤♠ ♦♥ ❆✳ ❲❡ ❤❛✈❡ t❤❡ ❜✐❣❣❡r ❝♠✳

✺ ■❢ ❛ ❜✐ ✐s s✐♥❣✉❧❛r✱ t❤❡♥ t❤❡r❡ ❛r❡ t✇♦ ❝❛s❡s✳ ■❢ r❛♥❦ ❜✐ = ✶✱

t❤❡♥ ✇❡ ❙❚❖P ✇✐t❤ t❤❡ ♦✉t♣✉t ❈ := ❜✐✳ ❊❧s❡✱ ✐❢ ✶ < r❛♥❦ ❜✐ < ♥✱ t❤❡♥ ✇❡ ❝♦♠♣✉t❡ t❤❡ t❤❡ r✐❣❤t ✐❞❡♥t✐t② ❡❧❡♠❡♥t ❡ ♦❢ A❜✐✱ s❡t A := ❡A❡ ❛♥❞ ❣♦ ❜❛❝❦ t♦ ❙t❡♣ ✶✳

✻ ❍❡r❡ |❜✐| ≥ √♥ ❢♦r ❡✈❡r② ✐✳ ●❡♥❡r❛t❡ ❈ = ♠

✐=✶ α✐❜✐✱ ✇❤❡r❡

α✐ ∈ Z✱ |α✐| ≤ ❝♠ ♥ |❜✐| ≤ ❝♠ √♥ ✉♥t✐❧ ❛ r❛♥❦ ✶ ❈ ✐s ❢♦✉♥❞✳ ❖✉t♣✉t t❤✐s ❈✳

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

❈♦rr❡❝t♥❡ss

■❢ ✇❡ ❤❛✈❡ s✉✣❝✐❡♥t ✭♣♦❧②♥♦♠✐❛❧✮ ♣r❡❝✐s✐♦♥ ❛t

✸ ✱ t❤❡♥ ❜②

❬❇✉❝❤♠❛♥♥ ✬✾✹❪ ❛❢t❡r

✹ ✇❡ ❤❛✈❡

|❜✶||❜✷| · · · |❜♠| ≤ (γ♠)

♠ ✷

✸ ✷ ♠ ✷

♠(♠−✶) ✷

. ❲❤❡♥ ❛t

✺ ✱ ✇❡ ❤❛✈❡ r❛♥❦ ❡ = ❦✱ t❤❡♥ ✐t ✐s ❡❛s② t♦ s❡❡ t❤❛t

❡A❡ ∼ = ▼❦(Q). ❆ r❛♥❦ ✶ ❡❧❡♠❡♥t ♦❢ ❡A❡ ❤❛s r❛♥❦ ✶ ✐♥ A ❛s ✇❡❧❧✳ ❚❤❡ t❤❡♦r❡♠ ❛♥❞ t❤❡ ❧❡♠♠❛ ✭❧❛tt❡r ❛♣♣❧✐❡❞ ❢♦r ✈ := ❈ ❛♥❞ |✈| ≤ ♥✮ s❤♦✇ t❤❛t ❛♥ ❡❧❡♠❡♥t ❈ ✇✐t❤ r❛♥❦ ♦♥❡ ❡①✐sts ❛t

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

❚✐♠✐♥❣

✶ r✉♥s ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛s ❛♥ ✛✲❛❧❣♦r✐t❤♠✳ ✷ ✱ ✹ ❛♥❞ ✺ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ❞❡t❡r♠✐♥✐st✐❝ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

❆t

✸ t❤❡ ♣r❡❝✐s✐♦♥ ♣❛r❛♠❡t❡r ✐s ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ✐♥♣✉t s✐③❡✱ ❤❡♥❝❡

❙❝❤ö♥❤❛❣❡✬s ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ r✉♥s ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❥✉♠♣s ❜❛❝❦ t♦

✶ ✐s ❜♦✉♥❞❡❞✱ ❤❡♥❝❡ ❡❛❝❤ ❙t❡♣ ✐s

❝❛rr✐❡❞ ♦✉t ✐♥ ❛ ❜♦✉♥❞❡❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳ ❋✐♥❛❧❧②✱ t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ❈ ❡♥✉♠❡r❛t❡❞ ❛t

✻ ✐s ❛t ♠♦st

(✷❝♠ √♥ + ✶)♠✳

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

❚❤❡ ❣❡♥❡r❛❧ ❝❛s❡

K ❜❡ ❛ ♥✉♠❜❡r ✜❡❧❞ ♦❢ ❞❡❣r❡❡ ❞ ♦✈❡r Q✱ t❤❡ ♠❛①✐♠❛❧ ♦r❞❡r ♦❢ K ✐s ❘✱ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ ❘ ✐s ∆✳ ▲❡t A ❜❡ ❛ ❝❡♥tr❛❧ s✐♠♣❧❡ ❛❧❣❡❜r❛ ♦✈❡r K s✉❝❤ t❤❛t A ∼ = ▼♥(K)✱ ❛♥❞ ❧❡t Λ ❜❡ ❛ ♠❛①✐♠❛❧ ♦r❞❡r ✐♥ A✳ Λ ✐s ✐s♦♠♦r♣❤✐❝ t♦ Λ′ :=      ❘ · · · ❘ ❏−✶ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❘ · · · ❘ ❏−✶ ❏ · · · ❏ ❘      , ✇❤❡r❡ ❏ ✐s ❛ ❢r❛❝t✐♦♥❛❧ ✐❞❡❛❧ ✐♥ K✳

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

▲❡t σ✶, . . . , σr ❜❡ t❤❡ ❡♠❜❡❞❞✐♥❣s ♦❢ K ✐♥t♦ R ❛♥❞ σr+✶, σr+✶, . . . , σr+s, σr+s ❜❡ t❤❡ ♥♦♥✲r❡❛❧ ❡♠❜❡❞❞✐♥❣s ♦❢ K ✐♥t♦ C❀ ✇❡ ❤❛✈❡ ❞ = r + ✷s✳ ❋♦r ✶ ≤ ✐ ≤ r + s ❝♦♥s✐❞❡r ❛♥ ❡♠❜❡❞❞✐♥❣ φ✐ ♦❢ A ✐♥t♦ ▼♥(C)✱ ✇❤✐❝❤ ❡①t❡♥❞s σ✐ ✭❢♦r ✐ ≤ r ✇❡ r❡q✉✐r❡ φ✐(A) ≤ ▼♥(R)✮✳ ❙✉❝❤ ❡♠❜❡❞❞✐♥❣s ❛r❡ ♣♦❧②✳ t✐♠❡ ❝♦♠♣✉t❛❜❧❡✳ ❙❡t ❜ = ✷ π ✷s♥ ∆♥ ✶

♥❞

= ✷ π ✷s

❞

∆

✶ ❞ . ▲❛❥♦s ❘ó♥②❛✐ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s

❙♠❛❧❧ ③❡r♦ ❞✐✈✐s♦rs ✐♥ Λ

❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐sts ❛ r❛♥❦ ♦♥❡ ❡❧❡♠❡♥t ① ∈ Λ s✉❝❤ t❤❛t t❤❡ ❡♥tr✐❡s ♦❢ t❤❡ ♠❛tr✐❝❡s φ✐(①) ❢♦r ✐ = ✶, . . . , s + r ❛❧❧ ❤❛✈❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ≤ ❜✳ ❚❤❡ ♣r♦♦❢ ✐s s✐♠✐❧❛r t♦ t❤❡ ❝❛s❡ K = Q✳ ❍❡r❡ ✇❡ ❡♠♣❧♦② ▼✐♥❦♦✇s❦✐✬s ❚❤♠ t♦ ❛ ❧❛tt✐❝❡ ✐♥ R♥❞✳ ❙♦♠❡ s♣❡❝✐❛❧ ❝❛s❡s✿ ✶✳ K = Q✱ ❘ = Z✱ t❤❡♥ ∆ = ✶✱ s = ✵✱ ❤❡♥❝❡ ❜ = ✶✳ ❲❡ ❤❛✈❡ ① ∈ Λ ✇❤✐❝❤ ❤❛s r❛♥❦ ✶ ❛s ❛ ♠❛tr✐① ❢r♦♠ ▼♥(Q)✱ ❛♥❞ ✇✐t❤ r❡s♣❡❝t t♦ A ֒ → ▼♥(R) ❤❛s ❡❧❡♠❡♥ts ♦❢ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❛t ♠♦st ✶✳ ✷✳ ■❢ ❉ > ✵ ✐s ❛ sq✉❛r❡❢r❡❡ ✐♥t❡❣❡r✱ K = Q( √ ❉)✱ t❤❡♥ ∆ = ❉✱ ✐❢ ❉ ✐s ❝♦♥❣r✉❡♥t t♦ ✶ ♠♦❞✉❧♦ ✹✱ ❛♥❞ ∆ = ✹❉✱ ✐❢ ❉ ✐s ❝♦♥❣r✉❡♥t t♦ ✸ ♠♦❞✉❧♦ ✹✳ ❚❤❡♥ s = ✵✱ ❞ = ✷✱ ❤❡♥❝❡ ❜ ≤ ✷ √ ❉✳

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

❆ ❣❡♥❡r❛❧ ✈❛r✐❛♥t ♦❢ ❋✐s❝❤❡r✬s ▲❡♠♠❛✳ ▲❡♠♠❛ ▲❡t ② ∈ Λ ❜❡ ❛♥ ❡❧❡♠❡♥t s✉❝❤ t❤❛t φ✐(②) ≤ √♥ ❤♦❧❞s ❢♦r ✐ = ✶, . . . , r + s✳ ❚❤❡♥ ② ✐s ❛ ③❡r♦ ❞✐✈✐s♦r ✐♥ A✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✇♦r❦s ✇✐t❤ ❛ r❡❞✉❝❡❞ ❜❛s✐s ♦❢ t❤❡ ❧❛tt✐❝❡ Γ ♦❢ ✈❡❝t♦rs (φ✶(②), . . . , φr(②), ℜ(φr+✶(②)), ℑ(φr+✶(②)), . . . , ℜ(φr+s(②)), ℑ(φr+s(②))), ✇❤❡r❡ ② ∈ Λ✳ ❲❡ ❤❛✈❡ Γ ⊂ R♥✷❞✳

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

❙❡t ♠ := ♥✷❞ ❆t t❤❡ ✜♥❛❧ st❡♣ ✇❡ ❤❛✈❡ ❛ r❡❞✉❝❡❞ ❜❛s✐s ❜✶, . . . , ❜♠ ♦❢ Γ ✇✐t❤ |❜✐| ≥ √♥ ❢♦r ❡✈❡r② ✐✳ ❲❡ ❣❡♥❡r❛t❡ ❛❧❧ ✐♥t❡❣❡r ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ✇ =

♠

• ✐=✶

α✐❜✐, ✇✐t❤ |α✐| ≤ ❝♠ ✷ π ✷s

❞

∆

✶ ❞

♥(r + s) ✉♥t✐❧ ❛ ✇ ✐s ❢♦✉♥❞ s✉❝❤ t❤❛t r❛♥❦ ① = ✶ ❤♦❧❞s ❢♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ① ∈ Λ✳

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

❆♣♣❧✐❝❛t✐♦♥s

❈♦r♦❧❧❛r② K ✐s ❛ ♥✉♠❜❡r ✜❡❧❞✱ A ✐s ❣✐✈❡♥ ❜② str✉❝t✉r❡ ❝♦♥st❛♥ts✱ ❛♥❞ A ∼ = ▼♥(K) ❢♦r ❛♥ ♥ > ✶✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ③❡r♦ ❞✐✈✐s♦r ① ∈ A ✇❤✐❝❤ ❛❞♠✐ts ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ ❝♦♦r❞✐♥❛t❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥♣✉t ❜❛s✐s ♦❢ A✳ ❙✉❝❤ ❛♥ ① ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣s♣❛❝❡✳ Pr♦♦❢✳ ▲❡t ❝✶, . . . , ❝♥✷ ❜❡ t❤❡ ❜❛s✐s ♦❢ Λ ❣✐✈❡♥ ❜② t❤❡ ■❘ ❛❧❣♦r✐t❤♠✳ ❊①♣r❡ss t❤❡ ❡❧❡♠❡♥t ① ♦❢ t❤❡ ❚❤❡♦r❡♠ ✐♥ t❤✐s ❜❛s✐s✿ ① = α✶❝✶ + α✷❝✷ + · · · + α♥✷❝♥✷ ✇✐t❤ α✐ ∈ Z✳ ❯s✐♥❣ t❤❛t ① ≤ ❜♥✱ ❛♥❞ t❤❛t t❤❡ ✈❡❝t♦rs ❝✐ ❤❛✈❡ ♣♦❧②♥♦♠✐❛❧ s✐③❡✱ ❈r❛♠❡r✬s r✉❧❡ ✐♠♣❧✐❡s ❛ ♣♦❧②♥♦♠✐❛❧ ❜♦✉♥❞ ♦♥ t❤❡ s✐③❡ ♦❢ t❤❡ α✐✳

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

◆♦r♠ ❡q✉❛t✐♦♥s

❚❤❡r❡ ✐s ❛♥ ✐♠♣♦rt❛♥t ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ s♣❧✐t ❝②❝❧✐❝ ❛❧❣❡❜r❛s ❛♥❞ r❡❧❛t✐✈❡ ♥♦r♠ ❡q✉❛t✐♦♥s✳ ❈♦r♦❧❧❛r② ▲❡t K ❜❡ ❛ ♥✉♠❜❡r ✜❡❧❞✱ L ❜❡ ❝②❝❧✐❝ ❡①t❡♥s✐♦♥ ♦❢ K✱ ❛♥❞ ❛ ∈ K✳ ■❢ t❤❡ ❛ ♥♦r♠ ❡q✉❛t✐♦♥ ◆L/K(①) = ❛ ✐s s♦❧✈❛❜❧❡✱ t❤❡♥ t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥ ✇❤♦s❡ st❛♥❞❛r❞ r❡♣r❡s❡♥t❛t✐♦♥ ❤❛s ♣♦❧②♥♦♠✐❛❧ s✐③❡ ✭✐♥ t❡r♠s ♦❢ t❤❡ s✐③❡ ♦❢ t❤❡ st❛♥❞❛r❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❛♥❞ ❛ ❜❛s✐s ♦❢ L✮✳ ❋✉rt❤❡r♠♦r❡✱ ❢♦r ✜①❡❞ K ❛♥❞ ✜①❡❞ ❞❡❣r❡❡ |L : K|✱ ❛ s♦❧✉t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✛✲❛❧❣♦r✐t❤♠✳

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

■s♦♠♦r♣❤✐s♠ ♦❢ ❝❡♥tr❛❧ s✐♠♣❧❡ ❛❧❣❡❜r❛s

❈♦r♦❧❧❛r② K ✐s ❛ ♥✉♠❜❡r ✜❡❧❞ ♦❢ ❞❡❣r❡❡ ❞ ❛♥❞ ❞✐s❝r✐♠✐♥❛♥t ∆ ♦✈❡r Q✳ ▲❡t A, B ❜❡ ✐s♦♠♦r♣❤✐❝ ❝❡♥tr❛❧ s✐♠♣❧❡ ❛❧❣❡❜r❛s ♦✈❡r K ♦❢ ❞✐♠❡♥s✐♦♥ ♥✷✱ ❣✐✈❡♥ ❜② str✉❝t✉r❡ ❝♦♥st❛♥ts✳ ❙✉♣♣♦s❡ t❤❛t ❞✱ ♥ ❛♥❞ |∆| ❛r❡ ❜♦✉♥❞❡❞✳ ❚❤❡♥ ❛♥ ✐s♦♠♦r♣❤✐s♠ A → B ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✛✲❛❧❣♦r✐t❤♠✳ ❚❤❡ r✉♥♥✐♥❣ t✐♠❡ ✐s ♣♦❧②♥♦♠✐❛❧ ✐♥ |∆| ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ ✐♥ ♥ ❛♥❞ ❞✳ Pr♦♦❢✳ ❲❡ ❤❛✈❡ A ∼ = B ✐❢ ❛♥❞ ♦♥❧② ✐❢ A ⊗K B♦♣ ∼ = ▼♥✷(K). ▲❡t ❱ ❜❡ ❛ s✐♠♣❧❡ A ⊗K B♦♣ ♠♦❞✉❧❡✳ ❚❤❡♥ ❞✐♠K❱ = ♥✷✳ ❙♦ ❱ ✱ ❛s ❧❡❢t A✲♠♦❞✉❧❡✱ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ r❡❣✉❧❛r ❧❡❢t A ♠♦❞✉❧❡✳ ❙✐♠✐❧❛r❧②✱ ❱ ✐s ❛ r❡❣✉❧❛r r✐❣❤t B♦♣✲♠♦❞✉❧❡✳ ▲❡t ✈ ∈ ❱ ❜❡ ❛ ❣❡♥❡r❛t✐♥❣ ❡❧❡♠❡♥t ♦❢ ❱ ❛s ❛♥ A✲♠♦❞✉❧❡✱ ❛♥❞ ❛❧s♦ ❛s ❛ B♦♣ ♠♦❞✉❧❡✳

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

❚❤❡♥ φ : ❛ → ❛✈ ✐s ❛ ❧❡❢t A✲♠♦❞✉❧❡ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ A t♦ ❱ ✳ ❆❧s♦✱ ψ : ❜ → ✈❜ ✐s ❛ r✐❣❤t B♦♣✲♠♦❞✉❧❡ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ B♦♣ t♦ ❱ ✳ ❚❤❡♥ σ = ψ−✶φ ✐s ❛ K ❛❧❣❡❜r❛ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ A ❛♥❞ B✳ ■t ✐s ❝❧❡❛r t❤❛t σ ✐s K ❧✐♥❡❛r✳ ❋♦r ❛ ∈ A✱ σ❛ ✐s t❤❡ ✉♥✐q✉❡ ❡❧❡♠❡♥t ❜ ∈ B ✇✐t❤ ❛✈ = ✈❜. ❚❤❡r❡❢♦r❡ σ(❛✶❛✷) ✐s t❤❡ ✉♥✐q✉❡ ❜ ∈ B ✇✐t❤ ❛✶❛✷✈ = ✈❜✳ ❇✉t ❛✶❛✷✈ = ❛✶✈(σ❛✷) = ✈(σ❛✶)(σ❛✷)✱ ✇❤❡♥❝❡ σ(❛✶❛✷) = (σ❛✶)(σ❛✷)✳

• ▲❛❥♦s ❘ó♥②❛✐

❥♦✐♥t ✇♦r❦ ✇✐t❤ ●á❜♦r ■✈❛♥②♦s✱ ➪❞á♠ ❉✳ ▲❡❧❦❡s✱ ❛♥❞ ❏♦s❡❢ ❙❝❤✐❝❤♦ ❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s

❚❡♥s♦r ♣r♦❞✉❝ts ♦❢ ❧❛tt✐❝❡s

▲❡t ▲ ⊂ R♠ ❛♥❞ ▼ ⊂ R♥ ❜❡ ❧❛tt✐❝❡s✳ ❲❡ ❤❛✈❡ ▲ ⊗Z ▼ ֒ → R♠ ⊗R R♥. ❚❤✐s ❛❧❧♦✇s ♦♥❡ t♦ ❞❡✜♥❡ t❤❡ ❧❛tt✐❝❡ ▲ ⊗ ▼✳ ▲⊗▼ ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ s❡t ♦❢ ♠ ❜② ♥ ♠❛tr✐❝❡s ♦✈❡r R ✇❤✐❝❤ ❛r❡ ✐♥t❡❣r❛❧ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❞②❛❞s ①②❚✱ ✇❤❡r❡ ① ∈ ▲ ❛♥❞ ② ∈ ▼✳ R♠ ⊗ R♥ ✐s ❛♥ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✇✐t❤ ①✶ ⊗ ②✶, ①✷ ⊗ ②✷ = ①✶, ①✷②✶, ②✷. ❚❤❡ ♥♦r♠ ♦♥ R♠ ⊗ R♥ ✐s t❤❡ ❋r♦❜❡♥✐✉s ♥♦r♠ ♦♥ ▼♠,♥(R)✳ ❲❡ ❤❛✈❡ Λ ∼ = P▼♥(Z)P−✶ ∼ = ◗Z♥ ⊗ (◗Z♥)∗.

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

■♠♣r♦✈❡♠❡♥t ✇❤❡♥ K = Q ❛♥❞ ♥ ≤ ✹✸

❘❡s✉❧ts ♦❢ ❨✳ ❑✐t❛♦❦❛ ✭✶✾✼✼✮ ♦♥ t❡♥s♦r ♣r♦❞✉❝ts ♦❢ ❧❛tt✐❝❡s ✐♠♣❧② t❤❛t t❤❡ s❤♦rt❡st ♥♦♥③❡r♦ ❡❧❡♠❡♥t ❈ ✐♥ t❤❡ r✐♥❣ P▼♥(Z)P−✶ ♠✉st ❤❛✈❡ r❛♥❦ ♦♥❡✱ ✇❤❡♥ P ✐s ❛♥ ✐♥✈❡rt✐❜❧❡ r❡❛❧ ♠❛tr✐①✱ ❛♥❞ ♥ ≤ ✹✸. ❆❜♦✈❡ ❢❛✐❧s ❜❛❞❧② ❢♦r ♥ ≥ ✷✾✷✳ ❑✐t❛♦❦❛✬s r❡s✉❧t ❧❡❛❞s t♦ ❛ s✐♠♣❧❡r ❛♥❞ ❢❛st❡r ♠❡t❤♦❞ ♦✈❡r Q✱ ✇❤❡♥ ♥ ≤ ✹✸✳

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

❆♥ ✐♠♣r♦✈❡❞ ❛❧❣♦r✐t❤♠ ❢♦r K = Q✱ ♥ ≤ ✹✸

✶ ❯s❡ t❤❡ ■❘ ❛❧❣♦r✐t❤♠ t♦ ❝♦♥str✉❝t ❛ ♠❛①✐♠❛❧ ♦r❞❡r Λ ✐♥ A✳ ✷ ❈♦♠♣✉t❡ ❛♥ A ֒

→ ▼♥(R)✳ ❯s❡ t❤❡ ❞❡r❛♥❞♦♠✐③❛t✐♦♥ ❢r♦♠ ❬❞❡

• r❛❛❢✱ ■✈❛♥②♦s ✬✵✵❪ ♦❢ ❬❊❜❡r❧② ✬✾✶❪✳ ❚❤✐s ❣✐✈❡s ❛ · ♦♥ A✳

✸ ❈♦♠♣✉t❡ ❛ s✉✣❝✐❡♥t❧② ♣r❡❝✐s❡ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❆ ♦❢ ♦✉r

❜❛s✐s ❇ ♦❢ Λ ❜② t❤❡ ♠❡t❤♦❞ ♦❢ ❬❙❝❤ö♥❤❛❣❡ ✬✽✷❪✳

✹ ❈♦♠♣✉t❡ ❛ r❡❞✉❝❡❞ ❜❛s✐s ❜✶, . . . , ❜♠ ♦❢ t❤❡ ❧❛tt✐❝❡ Λ ⊂ R♠ ❜②

t❤❡ ▲▲▲ ❛❧❣♦r✐t❤♠ ♦♥ ❆✳ ❲❡ ❤❛✈❡ t❤❡ ❜✐❣❣❡r ❝♠✳

✺ ●❡♥❡r❛t❡ ❈ = ♠

✐=✶ α✐❜✐✱ ✇❤❡r❡ α✐ ∈ Z✱

|α✐| ≤ ❝♠ ♠✐♥{γ′

♥, |❜✶|}

|❜✐| ≤ ❝♠ ✉♥t✐❧ ❛ r❛♥❦ ✶ ❈ ✐s ❢♦✉♥❞✳ ❖✉t♣✉t t❤✐s ❈✳

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

❆ ❑✐t❛♦❦❛ t②♣❡ ❜♦✉♥❞

❆ss✉♠❡ t❤❛t ✷ ≤ ♥ ≤ ✽✱ ❛♥❞ s❡t Λ = P▼♥(Z)P−✶✱ ✇❤❡r❡ P ✐s ❛♥ ✐♥✈❡rt✐❜❧❡ r❡❛❧ ♠❛tr✐①✳ ▲❡t ♠ ❜❡ t❤❡ ♠✐♥✐♠❛❧ ♥♦r♠ ♦❢ t❤❡ ♥♦♥③❡r♦ ❡❧❡♠❡♥ts ♦❢ Λ✳ ❚❤❡♦r❡♠ ▲❡t ❉ ∈ Λ ✇✐t❤ r❛♥❦ ❛t ❧❡❛st ✷✳ ❚❤❡♥ ❉ ≥

• ✸

✷♠✳

❆❧❧♦✇s t♦ r❡❞✉❝❡ ❢✉rt❤❡r t❤❡ s❡❛r❝❤ s♣❛❝❡ ❛t st❡♣

✺ ✳

❚❤❡ ♣r♦♦❢ ✐s ❛ r❡✜♥❡♠❡♥t ♦❢ ❑✐t❛♦❦❛✬s ❛r❣✉♠❡♥t✳

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

❚❤❡♦r❡♠ ▲❡t ▲ ❛♥❞ ▼ ❜❡ ❧❛tt✐❝❡s✳ ❋♦r ❡✈❡r② t❡♥s♦r ✈ ∈ ▲ ⊗ ▼ ♦❢ r❛♥❦ r ✇❡ ❤❛✈❡ ✈ ≥ r γ✷

r

λ✶(▲ ⊗ ▼). ▲❡♠♠❛ ▲❡t ❆, ❇ ∈ ▼♥(R) ❜❡ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ r❡❛❧ s②♠♠❡tr✐❝ ♠❛tr✐❝❡s✳ ❚❤❡♥ ❚r(❆❇) ≥ ♥

♥

√ ❞❡t ❆

♥

√ ❞❡t ❇✳ Pr♦♦❢ ♦❢ t❤❡ ❚❤❡♦r❡♠✳ ❲r✐t❡ ✈ ❛s ✈ =

r

• ✐=✶

①✐ ⊗ ②✐. ▲❡t ▲✶ ❜❡ t❤❡ ❧❛tt✐❝❡ ❣❡♥❡r❛t❡❞ ❜② {①✶, . . . , ①r}✱ ❛♥❞ ▼✶ ❜❡ t❤❡ ❧❛tt✐❝❡ s♣❛♥♥❡❞ ❜② {②✶, . . . , ②r}✳ ❚❤❡ r❛♥❦ ♦❢ t❤❡s❡ s✉❜❧❛tt✐❝❡s ✐s r✳

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

✈✷ = r

✐=✶ ①✐ ⊗ ②✐✷ = r ✐,❥=✶①✐, ①❥②✐, ②❥ =

❚r([①✐, ①❥]r

✐,❥=✶ · [②✐, ②❥]r ✐,❥=✶).

❇② t❤❡ ▲❡♠♠❛ ✈✷ ≥ r (❞❡t[①✐, ①❥] · ❞❡t[②✐, ②❥])✶/r . ◆♦✇ ❛ss✉♠❡ ❢♦r ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t ✈✷ < (r/γ✷

r )λ✶(▲ ⊗ ▼)✷✳

■t ❢♦❧❧♦✇s t❤❛t ✈✷ < r γ✷

r

(λ✶(▲)λ✶(▼))✷ ≤ r γ✷

r

(λ✶(▲✶)λ✶(▼✶))✷.

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

❈♦♠❜✐♥✐♥❣ t❤❡ t✇♦ ✐♥❡q✉❛❧✐t✐❡s r < r γ✷

r

· λ✶(▲✶)✷ (❞❡t[①✐, ①❥])✶/r · λ✶(▼✶)✷ (❞❡t[②✐, ②❥])✶/r ≤ r γ✷

r

γ✷

r = r,

❛s [①✐, ①❥]r

✐,❥=✶ ❛♥❞ [②✐, ②❥]r ✐,❥=✶ ❛r❡ ●r❛♠ ♠❛tr✐❝❡s ❢♦r ▲✶ ❛♥❞ ▼✶✱

r❡s♣❡❝t✐✈❡❧②✳ ❈♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❡ ♠✐♥✐♠✉♠ ♦❢ r/γ✷

✷ ❢♦r ✷ ≤ r ≤ ✽ ✐s ✸ ✷✱ ❛tt❛✐♥❡❞ ❛t r = ✷✳

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

■♠♣r♦✈❡♠❡♥t ❢♦r K = Q(

• −❞)✱ ♥ = ✷✱ ❞ = ✶ ♦r ✸

❆ss✉♠❡ t❤❛t A ∼ = ▼✷(K)✱ Λ ✐s ❛ ♠❛①✐♠❛❧ ♦r❞❡r ✐♥ A✳ ▲❡t φ ❜❡ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ A ✐♥t♦ ▼✷(C)✳ ❑✐t❛♦❦❛✲t②♣❡ r❡s✉❧ts ♦❢ ❘✳ ❈♦✉❧❛♥❣❡♦♥ ♦✈❡r Q( √ −❞) ❣✐✈❡✿ Pr♦♣♦s✐t✐♦♥ ❋♦r ❞ = ✶✱ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ s♠❛❧❧❡st ♥♦♥③❡r♦ ❡❧❡♠❡♥ts ♦❢ φ(Λ) ✇rt t❤❡ ❋r♦❜❡♥✐✉s ♥♦r♠ ❤❛s r❛♥❦ ♦♥❡✳ Pr♦♣♦s✐t✐♦♥ ❋♦r ❞ = ✸✱ t❤❡ s♠❛❧❧❡st ♥♦♥③❡r♦ ❡❧❡♠❡♥ts ♦❢ φ(Λ) ❤❛✈❡ r❛♥❦ ♦♥❡✳ ❆♣♣❧✐❝❛t✐♦♥✿ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ❉❡❧ P❡③③♦ s✉r❢❛❝❡s ♦❢ ❞❡❣r❡❡ ✽ ✭❲✳ ❆✳ ❞❡ ●r❛❛❢✱ ❏✳ Pí❧♥✐❦♦✈á✱ ❏✳ ❙❝❤✐❝❤♦✱ ✷✵✵✾✮✳

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

Pr♦❜❧❡♠s ❢♦r ❢✉t✉r❡ ✇♦r❦

■♥ t❤❡ ❝❛s❡ A ∼ = ▼♥(K) ❛❧❧♦✇ ♥ ♦r✴❛♥❞ K ✈❛r② ✭❡✈❡♥ ✇✐t❤ ❞ ✜①❡❞✮ ❙t✉❞② t❤❡ ❝❛s❡ ▼♥(D)✱ ✇❤❡r❡ D ✐s ❛ s❦❡✇✜❡❧❞ ♦✈❡r K ❉❡✈❡❧♦♣ ❛ ♣r❛❝t✐❝❛❧ ❛❧❣♦r✐t❤♠ ❊①t❡♥❞ t❤❡ ✐♠♣r♦✈❡♠❡♥t t♦ s♠❛❧❧ ♥✉♠❜❡r ✜❡❧❞s✱ s❛② ✐♠❛❣✐♥❛r② q✉❛❞r❛t✐❝ ✜❡❧❞s ✇✐t❤ s♠❛❧❧ ❞✐s❝r✐♠✐♥❛♥t

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

P❛♣❡rs✿

• ✳ ■✈❛♥②♦s✱ ▲✳ ❘✳✱ ❏✳ ❙❝❤✐❝❤♦✿

❙♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s ♦✈❡r ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡r ✜❡❧❞s✱ ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛✱ ✸✺✹✭✷✵✶✷✮✱ ✷✶✶✲✷✷✸✳ ❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✶✵✻✳✻✶✾✶

• ✳ ■✈❛♥②♦s✱ ➪✳ ❉✳ ▲❡❧❦❡s✱ ▲✳ ❘✳✿

■♠♣r♦✈❡❞ ❛❧❣♦r✐t❤♠s ❢♦r s♣❧✐tt✐♥❣ ❢✉❧❧ ♠❛tr✐① ❛❧❣❡❜r❛s✱ ❏P ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛✱ ◆✉♠❜❡r ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✷✽✭✷✵✶✸✮✱ ✶✹✶✕✶✺✻✳ ❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✷✶✶✳✶✸✺✻

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

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

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