- #1
JPanthon
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Homework Statement
Thanks to everyone who has helped me so far - I'm very grateful.
(1) Prove that the multiplicative inverse in any field is unique
(2) Prove the cancellation law | ab = ac => b=c
(3) Prove (-1)a = -a
Homework Equations
The field axioms: http://mathworld.wolfram.com/FieldAxioms.html
The Attempt at a Solution
(#1)
Let: a is an element of a field
Let: b and c are multiplicative inverses of a, such that, (1) ab = 1, and (2) ac = 1
Proof.
Start with (1)
ab = 1
(ab)/b = (1/b)
a (b/b)= (1/b)
a 1 = (1/b)
a = (1/b)
(2)
ac = 1
(1/b)c = 1
b((1/b)c)) = b(1)
(b/b)c = b
1c = b
c = b
QED
(#2)
Let: a, b are elements of F
Let: z is the mutliplicative inverse of x
Proof.
ax = bx
z(ax) = z(bx)
z(xa) = z (xb)
(zx)a = (zx)b
1a = 1b
a = b
QED
(#3)
Let: a is in a field
Let: -1 is the additive inverse of 1
Proof.
(-1)a = (-1 + 0) a
(-1)a = (-1 + (-1 + 1)) a
(-1)a = -1a + (-1a + 1a)
(-1)a = -1a + (1(-a + a))
(-1)a = -1a + (-a + a)
(-1)a = 1(-a) + (-a + a)
(-1)a = -a + (-a + a)
(-1)a = -a + (a + (-a))
(-1)a = (-a + a) + (-a)
(-1)a = 0 + (-a)
(-1)a = -a
QED
Thank you very much!
Please be as critical as possible, I really want to learn.
This is for a first year algebra class.
J.Anthony