Đại số trừu tượng và lý thuyết số

When division fails

• Modulo 8, $3 \cdot 3 = 9 \equiv 1$, so 3 has an inverse. But $2x \equiv 1 \pmod 8$ has no solution: every $2x$ is even.
• GRE algebra and number theory test which structure you are in — group, ring, integral domain 整环, field — and what that structure permits.

Groups

• A group needs closure, associativity, an identity and inverses. Commutativity is extra (abelian).
• $(\mathbb Z, +)$ is a group; $(\mathbb Z, \times)$ is not (2 has no inverse); $(\mathbb Q \setminus \{0\}, \times)$ is.

Homomorphisms and kernels 核

• $\varphi(ab) = \varphi(a)\varphi(b)$. The kernel is a normal subgroup, and $G/\ker\varphi \cong \operatorname{im}\varphi$.
• Reduction $\mathbb Z \to \mathbb Z_n$ has kernel $n\mathbb Z$, so $\mathbb Z / n\mathbb Z \cong \mathbb Z_n$. $\varphi$ is injective exactly when the kernel is trivial.

Rings, domains, fields

• Integral domain: commutative ring with $1$ and no zero divisors 零因子. Field: every nonzero element invertible.
• $\mathbb Z$ is a domain, not a field. $\mathbb Z_6$ has $2 \cdot 3 = 0$, so it is not a domain. $\mathbb Z_p$ ($p$ prime) is a field. Every finite domain is a field.

Congruences and inverses

• $a$ is invertible mod $n$ exactly when $\gcd(a, n) = 1$. Cancel a factor only if it is invertible.

Recap