{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-polynomial-ring-over-finite-commutative-ring"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-polynomial-ring-over-finite-commutative-ring-01","title":{"zh-Hant":"有限交換環上的多項式環：在 (ℤ/4ℤ)[x] 中做加法、卷積與次數核對","en":"Polynomial ring over a finite commutative ring: addition, convolution and degree checks in (ℤ/4ℤ)[x]"},"summary":{"zh-Hant":"在 R=ℤ/4ℤ 上取 f=1+2x、g=2x。依係數序列定義計算 f+g、f·g、f·f、g·g 並正規化；證明首項係數公式 (fg)_{m+n}=f_m g_n，說明為何此處 deg(fg)=deg f+deg g 失敗；最後證明 R[x] 不是有限環。","en":"Over R=ℤ/4ℤ take f=1+2x and g=2x. Using the coefficient-sequence definition, compute and normalize f+g, f·g, f·f and g·g; prove the top-coefficient formula (fg)_{m+n}=f_m g_n and explain why deg(fg)=deg f+deg g fails here; finally prove that R[x] is not a finite ring."},"mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"]}],"title":"概念推導導讀"}