MTH2001 Algebraic Structures Pre-Lecture 2

• Lecture Notes.
• Lecture Notes (Detailed).
• Lecture: MTH2001 Algebraic Structures Lecture 2.

Fields, Subfields, Quick Subfield Theorem

Maybe make a Venn Diagram for Algebraic Structures?

The final extension! Fields are just integral domains with inverses. This makes the algebraic structure hierarchy…

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

Formally…

Definition of a Field

A field is a commutative ring in which the set of non-zero elements form a group with respect to multiplication. I.e.,

A set $F$ with two operations $+$ and $\ast$ is called a field if:

• $(F,+)$ is an abelian group,
• $(F^{\ast },\cdot )$ is an abelian group.
• for all $x,y,z\in F$, distributivity holds.

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We can test and prove everything formally, but often another QST is used (quick subfield theorem/test):

Definition of Quick Subfield Theorem

Let $F$ be a field and $K$ a subset of $F$. Then $K$ is a subfield of $F$ if and only if:

1. $K$ contains the zero and the unity of F,
2. if $a,b\in L$ then $a+b$ and $ab$ belong to $K$,
3. if $a\in K$ then $-a\in K$,
4. if $a\in K$ and $a\ne 0$ then $a^{-1}\in K$.

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This can be used to show that, for example, Gaussian Integers are not a subfield of $\mathbb{C}$, but Gaussian Rationals are. We can also show that every finite integral domain is a field, which makes testing even easier (and can demonstrate other properties that fields have).

We can also show that things like integers modulo $n$ are fields if and only if $n$ is prime (as it makes the integral domain finite).

Homomorphisms

As previously defined, a group homomorphism is a map between groups that preserves group operations.

Similarly, rings also have homomorphisms where they preserve BOTH operations (since they have two). Hence,

Definition of a Ring Homomorphism

Let $R$ and $S$ be rings. A ring homomorphism from $R$ to $S$ is a mapping $\theta :R\to S$ that satisfies:

• $\theta (a+b)=\theta (a)+\theta (b)$, and
• $\theta (ab)=\theta (a)\theta (b)$, for all $a,b\in R$.

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Similar relationships between fields, integral domains, and rings, naturally exist with their homomorphisms, too.

Proving these homomorphism is as easy as direct proof.

Ring homomorphism properties:

If $\theta :R\to S$ is a ring homomorphism, then

• $\theta (0_R)=0_S$,
• $\theta (-a)=-\theta (a)$ for all $a\in R$.