84 lines
2.1 KiB
Plaintext
84 lines
2.1 KiB
Plaintext
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\section{FIELDS}
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1.12 Definition A field is a set $F$ with two operations,
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called addition and multiplication, which satisfy the
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following so-called "field axioms" (A), (M), and (D):
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\section{(A) Axioms for addition}
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(A1) If $x \in F$ and $y \in F$, then their sum $x+y$ is in $F$.
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(A2) Addition is commutative: $x+y=y+x$ for all $x, y \in F$.
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(A3) Addition is associative: $(x+y)+z=x+(y+z)$ for all $x, y, z \in F$.
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(A4) $F$ contains an element 0 such that $0+x=x$ for every $x \in F$.
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(A5) To every $x \in F$ corresponds an element $-x \in F$ such that
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$$
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x+(-x)=0
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$$
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(M) Axioms for multiplication
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(M1) If $x \in F$ and $y \in F$, then their product $x y$ is in $F$.
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(M2) Multiplication is commutative: $x y=y x$ for all $x, y \in F$.
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(M3) Multiplication is associative: $(x y) z=x(y z)$ for all $x, y, z \in F$.
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(M4) $F$ contains an element $1 \neq 0$ such that $1 x=x$ for every $x \in F$.
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(M5) If $x \in F$ and $x \neq 0$ then there exists an element $1 / x \in F$ such that
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$$
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x \cdot(1 / x)=1 \text {. }
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$$
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\section{(D) The distributive law}
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$$
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x(y+z)=x y+x z
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$$
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holds for all $x, y, z \in F$.
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https://leanprover-community.github.io/mathematics_in_lean/C02_Basics.html#
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-/
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class Field (α : Type) where
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add : α → α → α
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comm: ∀ x y : α, add x y = add y x
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ass: ∀ x y z : α, (add (add x y) z) = (add x (add x z))
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mul : α → α → α
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#check Field.mk
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#check Field Nat
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-- (A1) If $x \in F$ and $y \in F$, then their sum $x+y$ is in $F$.
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-- closure is not needed since add implies
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-- (A2) Addition is commutative: $x+y=y+x$ for all $x, y \in F$.
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-- since we are defining what the structure (class) is, then we define that it is commutative (then later you can show specifc objects like Q, Reals are Fields)
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-- Goal prove rationals are a field (to debug my definition of field) & use def of rationals
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-- structure == inductive, but structure has one constructor
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-- class ≈ structure + other machinery
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/-
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structure foo_s where
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x : Nat
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y : Nat
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#check foo_s.mk
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inductive foo_i
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| mk (x y : Nat)
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def foo_i.x : foo_i → Nat
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| mk x _ => x
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#check foo_s.x
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#check foo_i.x
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-/
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