Introduction to Abstract Algebra

:::note Resources

• https://www-users.cse.umn.edu/~brubaker/docs/152/152groups.pdf

:::

Informal comments

A GROUP is a set in which you can perform one operation with some nice properties. A RING is a set equipped with two operations, called addition and multiplication. A RING is a GROUP under addition and satisfies some of the properties of a group for multiplication. A FIELD is a GROUP under both addition and multiplication.

Group

Definition

A group is a set $\mathcal{G}$ (i.e. ) and satisfies:

1. Closure (called $\mathcal{G}$ is closed under the operation $\bullet$ )

$$\forall a, b \in \mathcal{G} \; a \bullet b \in \mathcal{G}$$

2. Identity

$$\exists e \in \mathcal{G} \; \mathrm{s.t.} \; \forall a \in \mathcal{G} \; e \bullet a = a \bullet e = a$$

3. Inverse ($e$ is the identity above)

$$\forall a \in \mathcal{G} \; \exists b \in \mathcal{G} \; \mathrm{s.t.} \; a \bullet b = b \bullet a = e$$

4. Associativity

$$\forall a, b, c \in \mathcal{G} \; a \bullet (b \bullet c) = (a \bullet b) \bullet c$$

Abelian Group: A group is said to be “abelian” if

$$\forall a, b \in G \; a \bullet b = b \bullet a$$

Examples

• $\mathbb{Z}/n\mathbb{Z}$ - fancy notation for $\mathrm{integers} \mod n$ under modulo addition.
• $(\mathbb{Z}/n\mathbb{Z})^{\times}$ - fancy notation for $\mathrm{integers} \mod n$ which are relatively prime to n under multiplication.
• $\mathbb{Z}$ - integers under addition. Groups don’t have to be finite.

Ring

Definition

A ring is a set $\mathcal{R}$ (i.e. ) and satisfies:

1. $\mathcal{R}$ is closed under two operations $+$ and $\times$, i.e.

$$\forall a, b \in \mathcal{R} \; a + b \in \mathcal{R} \\\textrm{and}\\ \forall a, b \in \mathcal{R} \; a \times b \in \mathcal{R}$$

2. $\mathcal{R}$ is an abelian group under $+$

3. Associativity of $\times$

$$\forall a, b, c \in \mathcal{R} \; a \times (b \times c) = (a \times b) \times c$$

4. Distributive properties

$$\forall a, b, c \in \mathcal{R} \; a \times (b + c) = (a \times b) + (a \times c) \\\textrm{and}\\ \forall a, b, c \in \mathcal{R} \; (b + c) \times a = (b \times a) + (c \times a)$$

Examples

• $\mathbb{Z}/n\mathbb{Z}$ under modulo addition and modulo multiplication.
• $\mathbb{Z}$ - integers under addition and multiplication.
• $\mathbb{Z}[x]$ - fancy notation for all polynomials with integer coefficientss under polynomial addition and polynomial multiplication.

Field

Definition

A field is a set $\mathcal{F}$ (i.e. ) and satisfies:

1. $\mathcal{F}$ is closed under two operations $+$ and $\times$, i.e.

$$\forall a, b \in \mathcal{F} \; a + b \in \mathcal{F} \\\textrm{and}\\ \forall a, b \in \mathcal{F} \; a \times b \in \mathcal{F}$$

2. $\mathcal{F}$ is an abelian group under $+$

3. $\mathcal{F} - {e}$ (e is the identity under $+$) is an abelian group under $\times$

Examples

• For a prime $p$, $\mathbb{Z}/p\mathbb{Z}$ is a field under modulo addition and modulo multiplication, since (sometimes $\mathbb{F}_{p}$ is used to emphasize this group under these operations)

  • $\mathbb{Z}/p\mathbb{Z}$ is a group under modulo addition
  • $(\mathbb{Z}/p\mathbb{Z}) - \{0\} = (\mathbb{Z}/p\mathbb{Z})^{\times}$ is a group under modulo multiplication

• $\mathbb{Q}$ (rational numbers), $\mathbb{R}$ (real numbers), $\mathbb{C}$ (complex numbers) are all infinite fields under the regular addition and multiplication.

Non Examples

• Note that $\mathbb{Z}$ (integers) are NOT a field under regular addition and multiplication.

• If $n$ is not a prime, then $\mathbb{Z}/n\mathbb{Z}$ is not a field, since $(\mathbb{Z}/n\mathbb{Z}) - \{0\} = (\mathbb{Z}/n\mathbb{Z})^{\times}$. There are, in general, lots of other elements than $0$ which are not relatively prime to $n$ and hence have no inverse under modulo multiplication.