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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
[Classical Geometry] How do you solve #4? 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)
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbffe82f9-2d9a-4748-a8e9-659533c98536%2F5e7444b3-4343-4dac-9044-348e80d3524a%2Fcg61vh_processed.png&w=3840&q=75)
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.
Transcribed Image Text:Q4. The first two sub-questions have been discussed in detail in lecture. For the third one, asking yourself what does equilateral mean, and how do we interpret this in the context of the
Cartesian plane? The remaining things are routine construction/computation.
Somehow, as you have probably noticed, even routine computation might be messy if the coordinates are arbitrary. Somehow following what we have explained in the previous week,
you should not worry about it anymore: the ERM will help you, so that you can move the points to any position at your configuration. After all equilateral triangle stays equilateral after
rigid motion. (Why? justify this in detail!)
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