Abstract algebra and number theory · 抽象代数与数论
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| field/fiːld/ | 域 | yù |
| homomorphism/ˈhɒməmɔːfɪzəm/ | 同态 | tóng tài |
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.
答案前的判断
- 非零元素可能没有乘法逆元。整数算术与域算术遵循不同的规则。
- 你的目标：运用群、子群、同态商结构。
Read the relationship
- A group needs closure, associativity, identity and inverses. Commutativity is an additional condition.
- Distinguish rings, integral domains and fields.
阅读关系
- 群需满足封闭性、结合律、单位元和逆元。交换性是附加条件。
- 区分环、整环与域。
The integers under addition form:
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.
运用定义规则
- 同态保持运算。其核是正规子群，并标识了映射到单位元的元素。
- 应用整除性、同余式及初等数论。
Which has an inverse modulo 9?
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.
检查条件
- 域允许对每个非零元素进行除法。整数构成环但不是域。
- 应用整除性、同余式及初等数论。
模 8 下，3 有逆元 3 因为 3·3=9≡1。但 2 无逆元因为 gcd(2,8)=2。模素数时，每个非零剩余类都有逆元；这构成了有限域。
The remainder when 17 is divided by 5 is ____.
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.
应用题目格式
- 处理模n的同余式。剩余类a具有乘法逆元当且仅当 gcd(a,n)=1。
- 应用整除性、同余式及初等数论。
在模运算方程中取消因子前，必须检查该因子是否可逆。
Which answer fits this case? · 哪个答案符合此案例？
Use groups, subgroups, homomorphisms and quotient structures · 使用群、子群、同态及商结构
Every nonzero residue modulo a composite number has an inverse.
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.
保持区别
- 同态 (homomorphism) — 保持相关运算的映射。
- 域 (field) — 一个交换环，其中每个非零元都有逆元。
- 使用群、子群、同态和商结构。
- 区分环、整环与域。
- 应用整除性、同余式及初等数论。
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.