09-240/Classnotes for Tuesday September 15

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The real numbers A set \mathbb R with two binary operators and two special elements 0, 1 \in \mathbb R s.t.

F1.\quad \forall a, b \in \mathbb R, a + b = b + a \mbox{ and } a \cdot b = b \cdot a
F2.\quad \forall a, b, c, (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)
\mbox{(So for any real numbers } a_1, a_2, ..., a_n, \mbox{ one can sum them in any order and achieve the same result.}
F3.\quad \forall a, a + 0 = a \mbox{ and } a \cdot 0 = 0 \mbox{ and } a \cdot 1 = a
F4.\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1
\mbox{So } a + (-a) = 0 \mbox{ and } a \cdot a^{-1} = 1
\mbox{(So } (a + b) \cdot (a - b) = a^2 - b^2)
\forall a, \exists x, x \cdot x = a \mbox{ or } a + x \cdot x = 0
Note: or means inclusive or in math.
F5.\quad (a + b) \cdot c = a \cdot c + b \cdot c

Definition: A field is a set F with two binary operators \,\!+: F×FF, \times\,\!: F×FF and two elements 0, 1 \in \mathbb R s.t.

F1\quad \mbox{Commutativity } a + b = b + a \mbox{ and } a \cdot b = b \cdot a \forall a, b \in F
F2\quad \mbox{Associativity } (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)
F3\quad a + 0 = a, a \cdot 1 = a
F4\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1
F5\quad \mbox{Distributivity } (a + b) \cdot c = a \cdot c + b \cdot c

Examples

  1. F = \mathbb R
  2. F = \mathbb Q
  3. \mathbb C = \{ a + bi : a, b \in \mathbb R \}
    i = \sqrt{-1}
    \,\!(a + bi) + (c + di) = (a + c) + (b + d)i
    \,\!0 = 0 + 0i, 1 = 1 + 0i
  4. \,\!F_2 = \{ 0, 1 \}
  5. \,\!F_7 = \{ 0, 1,2,3,4,5,6 \}
  6. \,\!F_6 = \{ 0, 1,2,3,4,5 \} is not a field (counterexample)

Theorem: \,\!F_P for p>1 is a field IFF p is a prime number

Tedious Theorem

  1. a + b = c + d \Rightarrow a = c "cancellation property"
  2.  a \cdot b = c \cdot b , (b \ne 0) \Rightarrow a = c

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