추상 대수 및 수론
도입됨| English | 의미 |
|---|---|
| field/fiːld/ | 체 |
| homomorphism/ˈhɒməmɔːfɪzəm/ | 동형사상 |
A decision before an answer
- A nonzero element may have no multiplicative inverse. Integer arithmetic and field arithmetic obey different rules.
- Your goal: Use groups, subgroups, homomorphisms and quotient structures.
Read the relationship
- A group needs closure, associativity, identity and inverses. Commutativity is an additional condition.
- Distinguish rings, integral domains and fields.
The integers under addition form:
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Addition has identity 0 and inverse −a.
Use the defining rule
- A homomorphism preserves the operation. Its kernel is a normal subgroup and identifies elements mapping to the identity.
- Apply divisibility, congruences and elementary number theory.
Which has an inverse modulo 9?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
gcd(2,9)=1; its inverse is 5.
Check the conditions
- A field permits division by every nonzero element. Integers form a ring but not a field.
- Apply divisibility, congruences and elementary number theory.
Modulo 8, 3 has inverse 3 because 3·3=9≡1. But 2 has no inverse because gcd(2,8)=2. Modulo a prime, every nonzero residue has an inverse; this gives a finite field.
The remainder when 17 is divided by 5 is ____.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
17=3×5+2.
Apply the task format
- Work with congruences modulo n. A residue a has a multiplicative inverse exactly when gcd(a,n)=1.
- Apply divisibility, congruences and elementary number theory.
Do not cancel a factor in a modular equation without checking that it is invertible.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
군, 부분군, 동형 사상 및 quotient 구조 활용하기
Every nonzero residue modulo a composite number has an inverse.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
A residue sharing a factor with the modulus is not invertible.
Keep the distinctions
- homomorphism 同态 — A map preserving the relevant operation.
- field 域 — A commutative ring where each nonzero element has an inverse.
- Use groups, subgroups, homomorphisms and quotient structures.
- Distinguish rings, integral domains and fields.
- Apply divisibility, congruences and elementary number theory.
Match each term with its precise meaning in this lesson.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
난이도 3/5 · 200 코인 + 오늘 첫 완료 시 소액 보너스
The order follows the stated process; check each stage before the next.