5. Show the congruence axiom C3 for II (don't assume the field is Pythagorean)

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Chapter2: Second-order Linear Odes
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[Classical Geometry] How do you solve #5? The second picture is a hint (you don't need to solve the bullet points in the hint, just the asked question in the list of seven)

Q5. If the field is Pythagorean, I guess most guys will be super happy -- after it is explained in lecture in detail why C3 is true in a Cartesian field over a Pythagorean field. Now you no
longer have that condition. But wait, what you need is probably, technically at some key argument in your proof, an existence of a square root. If no Pythagorean property is available,
why don't we just look at the DEFINITION what is a square root? After all, the square root is defined for any field, not necessarily Pythagorean. But you got to be careful as √a = b
implies that b > 0(Check the definition of this symbol, which we discussed in lecture). Now you need to keep consistency w.r.t. the betweeness in your setting.
Transcribed Image Text:Q5. If the field is Pythagorean, I guess most guys will be super happy -- after it is explained in lecture in detail why C3 is true in a Cartesian field over a Pythagorean field. Now you no longer have that condition. But wait, what you need is probably, technically at some key argument in your proof, an existence of a square root. If no Pythagorean property is available, why don't we just look at the DEFINITION what is a square root? After all, the square root is defined for any field, not necessarily Pythagorean. But you got to be careful as √a = b implies that b > 0(Check the definition of this symbol, which we discussed in lecture). Now you need to keep consistency w.r.t. the betweeness in your setting.
1. Construct √an – bn using the ruler and compass, where a > b> 0 in an ordered field F
(assuming you have two points (0, 0) and (1,0), as always.)
2. Show that Q√3 = {a +b√3|a, b € Q} is a field by verifying the field axioms one by one.
3. Show that if (F, P) is an ordered field and a € F is such that a > 0, so is a ¹.
4. Let II be an ordered field.
(a). Explain what does 3 mean in F.
(b). Give an example of ordered field F where 3 does not have a square root in F
(c). Show that there is an equilateral triangle in II if and only if √3 € F.
5. Show the congruence axiom C3 for IIF (don't assume the field is Pythagorean)
6. Can we discuss incidence in IIF31? Can we discuss betweenness in IIF3₁1? Explain your
answer.
7. Consider the vector space F³ = {(x, y, z)|x, y, z € F}, where F is field.
(a). let ~ be a relation on F³ \ {0, 0, 0}, defined by
a = (x1, y₁, 21) ~ B = (x2, Y2, 22)
if there exists some > € K s.t. Aa= B. Show that A is an equivalence relation.
(b). Describe the equivalent classes [a] in F3.
(c). The projective plane over F, denoted by FP2 is the set of equivalent classes [a]. We
interpret the primitive term as
• point: [a], a € F³.
● line: {[a] a = (x, y, z), ax + by + cz = 0} for some fixed a, b, c = F.
(d). show that the line is well defined: if ax+by+cz = 0 holds for some a = = (x, y, z),
then it holds for any B = (x1, 9₁, 2₁) s.t. B € [a].
(e). How many points are there in F₂P²? List them.
(f). How many lines are there in F₂P2? List them.
(g). Show that F₂P2 is isomorphic to the Fano plane.
Transcribed Image Text:1. Construct √an – bn using the ruler and compass, where a > b> 0 in an ordered field F (assuming you have two points (0, 0) and (1,0), as always.) 2. Show that Q√3 = {a +b√3|a, b € Q} is a field by verifying the field axioms one by one. 3. Show that if (F, P) is an ordered field and a € F is such that a > 0, so is a ¹. 4. Let II be an ordered field. (a). Explain what does 3 mean in F. (b). Give an example of ordered field F where 3 does not have a square root in F (c). Show that there is an equilateral triangle in II if and only if √3 € F. 5. Show the congruence axiom C3 for IIF (don't assume the field is Pythagorean) 6. Can we discuss incidence in IIF31? Can we discuss betweenness in IIF3₁1? Explain your answer. 7. Consider the vector space F³ = {(x, y, z)|x, y, z € F}, where F is field. (a). let ~ be a relation on F³ \ {0, 0, 0}, defined by a = (x1, y₁, 21) ~ B = (x2, Y2, 22) if there exists some > € K s.t. Aa= B. Show that A is an equivalence relation. (b). Describe the equivalent classes [a] in F3. (c). The projective plane over F, denoted by FP2 is the set of equivalent classes [a]. We interpret the primitive term as • point: [a], a € F³. ● line: {[a] a = (x, y, z), ax + by + cz = 0} for some fixed a, b, c = F. (d). show that the line is well defined: if ax+by+cz = 0 holds for some a = = (x, y, z), then it holds for any B = (x1, 9₁, 2₁) s.t. B € [a]. (e). How many points are there in F₂P²? List them. (f). How many lines are there in F₂P2? List them. (g). Show that F₂P2 is isomorphic to the Fano plane.
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