Constructible Numbers, Fields and Surds
Original Notes adopted from February 5, 2002 (Week 18)
© P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong
Constructible Numbers
If a,b,c are constructible & > 0,
if b<c
c
= x
b
a
x = ac/b
if b> c
b
= a
c
x
x = ac/b
So can construct ac/b for a,b,c positive constructed numbers. In particular, take b =1, shows can construct
the product of any two constructible positive numbers.
Take c =1, show can construct quotient of any two constructible positive numbers.
Let C = set of all constructible numbers.
If x
∈
C, x
∈
C.
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∈
C
If a,b,
∈
C, so is a +b.
Definition:
A subset F of R is a number field if
1) 0,1
∈
F
2) If x,y
∈
F, so are x + y & x * y.
3) If x
∈
F, so is –x.
4) If x
∈
F & x
≠
0, then 1/x
∈
F.
Above we showed: C is a number field.
C
⊃
Q
Eg. R,Q are number fields
Q(
√
2) is defined to be {a + b
√
2: a,b
∈
Q}
Obviously Properties 1,3,4 hold
Property 2:
(a + b
¥
2 )(c + d
¥
2 ) = ac + 2bd + (bc + ad)
¥
2
∈
Q(
¥
2)
Property 4:
1
* ab
¥
2
=
ab
¥
2
=
a
+
b
¥
2
a +b
¥
2
ab
¥
2
a
2
– 2b
2
a
2
– 2b
2
a
2
– 2b
2
If a
2
– 2b
2
= 0
a
2
– 2b
2
(a/b)
2
= 2
⇒
¥
2 rational, contradiction.
∴
If a,b not both 0, 1/a+b
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 Spring '10
 Applebaugh
 Math, Rational number, Nth root, Algebraic number field, Quadratic field

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