Download Geometry over fields

Geometry over fields
A BSTRACT. This notes provide a short summary of the main results contained in
Chapter 3 and Section 21 of the book Geometry: Euclid and Beyond, by Robin
Hartshorne.
1. The Cartesian plane over a field
D EFINITION 1.1. A field is a set F with two operations + and · such that
(1) ( F, +) is an abelian group; we denote by 0 its identity element.
(2) Setting F ∗ = F \ {0}, then ( F ∗ , ·) is an abelian group; we denote by 1 its
identity element.
(3) The distributiva law holds:
∀ a, b, c ∈ F.
a(b + c) = ab + ac
D EFINITION 1.2. The cartesian plane over the field F, denoted by Π F is the set F 2
of ordered pairs of elements of F, which are called the points of Π F . A line is a
subset of Π F defined by a linear equation ax + by + c = 0, with a, b, c ∈ F such that
( a, b) 6= (0, 0). If b 6= 0 the line can be written as y = mx + q, and m is called the
slope of the line. If b = 0, then we say that the slope of the line is ∞.
P ROPOSITION 1.3. For any field F, the cartesian plane Π F satisfies Hilbert’s axioms (I1),
(I2), (I3) and (P).
T HEOREM 1.4 (Pappus). Let `, m be lines in Π F , with points A, B, C ∈ ` and A0 , B0 , C 0 ∈
m such that AC 0 k A0 C and BC 0 k B0 C. Then also AB0 k A0 B.
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2
P ROOF. Assume that ` and `0 meet at the point O (do the case ` k `0 as an
exercise). Up to a linear change of coordinates we can assume that O = (0, 0),
` = { x = 0}, `0 = { y = 0}, A = (0, 1), B0 = (1, 0).
Let C be the point (0, a) and C 0 be the point (b, 0).
Then the line AC 0 has equation y = −b−1 x + 1, therefore the line A0 C has equation
y = −b−1 x + a and A0 = (ba, 0).
The line B0 C has equation y = − ax + a, therefore the line BC 0 has equation y =
− ax + ab and B = (0, ab). The line AB0 has equation y = − x + 1; the parallel line
by A0 has equation y = − x + ba, hence it passes by B0 since ab = ba.
R EMARK 1.5. If F is just a skew-field, i.e. multiplication is not commutative, it is
still possible to define Π F . The above proof shows that Pappus Theorem holds in
Π F if and only if F is a field.
2. Ordered fields and betweenness
D EFINITION 2.1. An ordered field ( F, P) is a pair where F is a field, and P ⊂ F called the subset of positive elements, satisfies:
(1) If a, b ∈ P, then a + b ∈ P and ab ∈ P.
(2) ∀ a ∈ F one and only one of the following holds: a ∈ P, a = 0, − a ∈ P.
R EMARK 2.2. It is easy to show that, if ( F, P) is an ordered field, then the relation
defined as a > b if and only if a − b ∈ P is a strict total order on F. Clearly a ∈ P if
and only if a > 0.
P ROPOSITION 2.3 (15.3). If F is a field, and if there is a notion of betweenness in the
Cartesian plane Π F satisfying Hilbert’s axioms (B1)-(B4), then F is an ordered field. Conversely, if ( F, P) is an ordered field, we can define betweenness in Π F so as to satisfy
(B1)-(B4).
The idea of the proof is the following: if Π F has a notion of betweenness as above
we define P ⊂ F to be the set
{ a ∈ F | ( a, 0) is on the same side of (0, 0) as (1, 0) in the line y = 0}.
Conversely, if F is an ordered field, given A = ( a1 , a2 ), B = (b1 , b2 ) and C =
(c1 , c2 ) then A ∗ B ∗ C if one of the following holds:
• a1 < b1 < c1 ;
• a1 > b1 > c1 ;
• a1 = b1 = c1 and a2 < b2 < c2 or a2 > b2 > c2 .
GEOMETRY OVER FIELDS
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3. Congruences of segments and angles
Given an ordered field ( F, P) and points A = ( a1 , a2 ), B = (b1 , b2 ) in Π F we can
define
dist2 ( A, B) = ( a1 − b1 )2 + ( a2 − b2 )2 ,
and thus a relation of congruence of segments:
AB ∼
= CD
if and only if
dist2 ( A, B) = dist2 (C, D ).
P ROPOSITION 3.1. The relation of congruence of segments just defined satisfies Hilbert’s
axioms (C2)-(C3).
Given an angle α, defined by two rays r, r0 with a common point, lying on lines
`, `0 , of slopes m, m0 , we say that α is a right angle if either mm0 = −1 or, up
to exchange the lines, (m, m0 ) = (0, ∞). We say that α is an acute angle if it is
contained in the interior of a right angle, an obtuse angle if it contains a right angle
in its interior.
We define the tangent of α in the following way:



∞









 m0 − m tan α := 1 + mm0 







 m0 − m 

− 1 + mm0 with the conventions
∞−m
1
=
1+∞·m
m
if α is a right angle
if α is an acute angle
if α is an obtuse angle
m0 − ∞
1
= − 0.
0
1+∞·m
m
We say that two angles α and β are congruent if and only if tan α = tan β.
P ROPOSITION 3.2. The relation of congruence of angles just defined satisfies Hilbert’s
axioms (C4)-(C5).
3.1. Pytagorean fields.
D EFINITION 3.3. A field F is called Pythagorean if for every a ∈ F there exists b ∈ F
such that b2 = 1 + a2 .
E XAMPLE 3.4. The field Q of rational numbers is not Pythagorean.
P ROPOSITION 3.5 (16.1 - 17.2 - 17.1). If F is a Pythagorean, ordered field then the relations of congruence of segments and angles defined above in Π F satisfy Hilbert’s axioms
(C1)-(C6).
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C OROLLARY 3.6. If F is a Pythagorean, ordered field then Π F is an Hilbert plane satisfying (P).
D EFINITION 3.7. Let Π and Π0 be two Hilbert planes. An isomorphism between Π
and Π0 is a bijection ϕ : Π → Π0 compatible with the undefined notions, i.e.:
(1) A subset ` ⊂ Π is a line if and only if ϕ(`) ⊂ Π0 is a line.
(2) Three points A, B, C ∈ Π satisfy A ∗ B ∗ C if and only ϕ( A) ∗ ϕ( B) ∗ ϕ(C )
holds in Π0 .
(3) The line segments AB and CD are congruent in Π if and only if the line
segments ϕ( A)ϕ( B) and ϕ(C )ϕ( D ) are congruent in Π0 .
(4) Two angles α and β in Π are congruent if and only if the angles ϕ(α ) and
ϕ(β) are congruent in Π0 .
T HEOREM 3.8 (21.1). Let Π be a Hilbert plane satisfying the parallel axiom (P). Let F
be the ordered field of segment arithmetic in Π. Then F is a Pythagorean field, and Π is
isomorphic to the Cartesian plane Π F over the field F.
3.2. Euclidean fields and planes.
D EFINITION 3.9. An ordered field F is called Euclidean if for every a ∈ F there
exists b ∈ F such that b2 = a.
R EMARK 3.10. Clearly, an Euclidean field is also a Pythagorean field. The converse
does not hold, but there are no easy counterexamples.
P ROPOSITION 3.11 (16.2). If F is an ordered field and Π F is the Cartesian plane over F
then the following are equivalent:
(1) F is Euclidean;
(2) Π F satisfies the circle-circle intersection property (E);
(3) Π F satisfies the line-circle intersection property (LCI) (see Proposition 11.6).
D EFINITION 3.12. An Euclidean plane is an Hilbert plane satisfying (P) and (E).
C OROLLARY 3.13. If F is an ordered Euclidean field and Π F is the Cartesian plane over
F then Π F is an Euclidean plane.
T HEOREM 3.14 (10.4 - 11.8 - 12.1 -12.3 -12.4 - 12.5). Euclid’s propositions contained in
Books I-IV are valid in an Euclidean plane.
3.3. Archimedean fields.
D EFINITION 3.15. An ordered field F is called Archimedean if, for every a ∈ F there
exists n ∈ N such that n · 1 > a.
GEOMETRY OVER FIELDS
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D EFINITION 3.16. An Hilbert plane Π satisfies (A) if, given two segments AB and
CD, there exists n ∈ N such that nAB > CD (here nAB is a segment obtained by
adding n copies of the segment AB).
It can be shown that
P ROPOSITION 3.17 (15.4 -15.5). An ordered field is Archimedean if and only if Π F , the
cartesian plane over F, satisfies (A). Moreover, in this case, F is isomorphic (as an ordered
field) to a subfield of R.
E XAMPLE 3.18. Let us give an example of a non-Archimedean field. Set F := R(t), the
quotient field of R[t], the ring of polynomials in one variable with real coefficients. An
element ϕ of F can be written as a quotient of two polynomials f (t), g(t), with g(t) 6≡ 0;
considering it as a function ϕ(t), defined over R minus the zeroes of g, we can define
P ⊂ F as follows:
ϕ∈P
⇐⇒
∃ x0 ∈ R t.c. ϕ(t) > 0 ∀t > x0 .
It is straightforward to verify that P satisfies properties (1) and (2) in Definition 2.1, so it
defines an order in F.
Tale now ϕ(t) = t; it is clear that t − n ∈ P for every n ∈ N, so t > n
∀n ∈ N, hence
F is not Archimedean.
4. Rigid motions and SAS
D EFINITION 4.1. Let Π be a geometry consisting of the undefined notions of point,
line, betweenness, and congruence of line segments and angles, which may or may
not satisfy various of Hilbert’s axioms. A rigid motion of Π is a bijection ϕ : Π → Π
such that:
(1) ϕ sends lines into lines.
(2) ϕ preserves betweenness of collinear points.
(3) For any two points A, B we have AB ' ϕ( A)ϕ( B).
(4) For any angle α, we have α ' ϕ(α ).
It is clear that the composition of two rigid motions is again a rigid motion, and it
is not difficult to show that the inverse of a rigid motion is again a rigid motion,
hence the set of rigid motions of Π is a group with respect to composition.
Let Π be a geometry consisting of the undefined notions of point, line, betweenness, and congruence of line segments and angles. Then Π satisfies (ERM) if
(1) For any two points A, A0 ∈ Π, there is a rigid motion ϕ s.t. ϕ( A) = A0 .
(2) For any three points O, A, A0 , there is a rigid motion ϕ s.t. ϕ(O) = O and
−→
−→
ϕ sends the ray OA to the ray OA0 .
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(3) For any line `, there is a rigid motion ϕ s.t. ϕ(P) = P for all P ∈ ` and ϕ
interchanges the two sides of `.
It can be shown that
P ROPOSITION 4.2 (17.1, 17.4). Let Π be a geometry satisfying (I1)-(I3), (B1)-(B4), (C1)(C5). Then Π satisfies (C6) if and only if it satisfies (ERM).
P ROOF. We saw in the classroom this proof, which can be found on the textbook, combining the proofs of Propositions 17.1 and 17.4.