Complex Numbers and Field Theory

Practice flashcards covering the construction of complex numbers, field and group definitions, and the specific algebraic properties of R^2 and C from the lecture transcript.

$\mathbb{R}$ (The set of real numbers)

A complete ordered field, through which every ordered field is isomorphic.

Defective property of $\mathbb{R}$

The existence of polynomials of positive degree that cannot be written in the form $(x-a)(x-b)$, such as $x^2+1$.

Polynomial factorization property

Every polynomial of positive degree can be written as the product of polynomials of degree 1 or 2.

Factorization of $x^3-8$ (per transcript)

$$(x-2)(x^2+2x+2)$$

Factorization of $x^4+16$

$$(x-2)(x+2)(x^2+4)$$

Factorization of $x^4+1$

$(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)$.

Field (Definition)

A set $F$ with two binary operations $+$ and $\cdot$ such that $(F, +)$ is an abelian group, $(F \setminus \{0\}, \cdot)$ is an abelian group, and distributive laws $a(b+c) = ab+ac$ and $(a+b)c = ac+bc$ hold.

Group (Definition)

A set $G$ with a binary operation that is associative, contains an identity element $e$ ($ae=a=ea$), and for each $a \in G$ contains an inverse ($ab=e=ba$).

$\mathbb{R}^2$ (The set of ordered pairs)

The set $\mathbb{R} \times \mathbb{R} = \{(x,y)|x,y \in \mathbb{R}\}$ where $a = (a_1, a_2)$; $a_1$ is the first component and $a_2$ is the second component.

Addition in $\mathbb{R}^2$

$a+b = (a_1+b_1, a_2+b_2)$, which forms an abelian group.

Multiplication in $\mathbb{R}^2$

$ab = (a_1b_1 - a_2b_2, a_1b_2 + a_2b_1)$.

Relationship between $\mathbb{R}^2$ and Complex Numbers

$\mathbb{R}^2$ is not the field of complex numbers because it does not contain $\mathbb{R}$ as a subfield; it must be replaced with an isomorphic field to finish construction.

$\mathbb{C}$ (The field of complex numbers)

The set $\mathbb{C} = \{a+bi | a, b \in \mathbb{R}\}$ with operations defined for addition and multiplication.

Equality in $\mathbb{C}$

$a+bi = c+di \iff a=c$ and $b=d$ for $a, b, c, d \in \mathbb{R}$.

Multiplicative Identity of $\mathbb{R}^2 \setminus \{0\}$

The ordered pair $(1, 0)$.

Multiplicative Inverse in $\mathbb{R}^2 \setminus \{0\}$

For $a = (a_1, a_2)$, the inverse is $(\frac{a_1}{a_1^2+a_2^2}, -\frac{a_2}{a_1^2+a_2^2})$.

$$i^2$$

$-1$, where $i$ is the square root of $-1$ and $i \notin \mathbb{R}$.

Multiplicative Group of $\mathbb{C}$

Denoted by $\mathbb{C}^*$, it consists of $( \mathbb{C} \setminus \{0\}, \cdot )$, which is an abelian group.

Additive Group of $\mathbb{C}$

Denoted by $\mathbb{C}^+$, it consists of $( \mathbb{C}, + )$, which is an abelian group.