MTH2001 Algebraic Structures Lecture 2

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What defines a field in algebraic structures?

How do fields relate to integral domains and rings?

A field is an algebraic structure that satisfies:

1. Commutative ring: Addition and multiplication are commutative.
2. Multiplicative inverses: Every non-zero element has an inverse.
3. Integral domain: No zero divisors (i.e., $ab=0⟹a=0$ or $b=0$).

This hierarchy places fields as the most restrictive:

$$\text{Fields}\subset \text{Integral domains}\subset \text{Rings}.$$

Key Insight: Fields are integral domains where every non-zero element is a unit (invertible).

How do we identify subfields efficiently?

What is the Quick Subfield Theorem (QST)?

A subset $S\subseteq F$ of a field $F$ is a subfield if:

1. Closure under operations: $a\pm b,ab,a^{-1}\in S$ for all $a,b\in S$ ($b\ne 0$).
2. Non-empty: $S$ contains at least two elements (typically $0$ and $1$).

Example:

• Gaussian Integers $\mathbb{Z}[i]$ are not a subfield of $\mathbb{C}$ (e.g., $i^{-1}=-i\notin \mathbb{Z}[i]$).
• Gaussian Rationals $\mathbb{Q}(i)$ are a subfield (closed under inverses and operations).

What makes finite integral domains fields?

Every finite integral domain is a field.

Proof Sketch:

• In a finite set, injectivity of multiplication by a non-zero element implies surjectivity (pigeonhole principle).
• Hence, every non-zero element has an inverse.

Example:

• $\mathbb{Z}/p\mathbb{Z}$ (integers modulo prime $p$) is a field.
• $\mathbb{Z}/n\mathbb{Z}$ is a field if and only if $n$ is prime.

What are ring homomorphisms and their properties?

How do ring homomorphisms differ from group homomorphisms?

A ring homomorphism $\theta :R\to S$ preserves both ring operations: 3. Addition: $\theta (a+b)=\theta (a)+\theta (b)$. 4. Multiplication: $\theta (ab)=\theta (a)\theta (b)$. 5. Multiplicative identity: $\theta (1_R)=1_S$ (if rings are unital).

What fundamental properties do they inherit?

For any ring homomorphism $\theta$:

• $\theta (0_R)=0_S$.
• $\theta (-a)=-\theta (a)$.
• Kernel: $\ker (\theta )=\{a\in R:\theta (a)=0_S\}$ is an ideal of $R$.

Example: The map $\theta :\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ defined by $\theta (k)=k\mathrm{mod}n$ is a ring homomorphism with kernel $n\mathbb{Z}$.

Key Takeaways:

• Fields are maximal in the algebraic hierarchy, requiring inverses for all non-zero elements.
• The QST simplifies subfield verification by checking closure and element existence.
• Ring homomorphisms preserve structure across both operations, enabling transfer of algebraic properties.