[Classical Geometry] How do you solve #3? 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 youranswer.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 bya = (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]. Weinterpret 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. Q3. To figure out this question, you simply follow the definition.• What do we mean exactly by saying that a 0?What do we mean exactly by saying that a-¹ 0?The remaining thing is just an easy application of the axioms of ordered fields.

Question

[Classical Geometry] How do you solve #3? 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 youranswer.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 bya = (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]. Weinterpret 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. Q3. To figure out this question, you simply follow the definition.• What do we mean exactly by saying that a > 0?What do we mean exactly by saying that a-¹ > 0?The remaining thing is just an easy application of the axioms of ordered fields.

Accepted Answer

The details solution of the given question is given below in step 2.

3. Show that if (F, P) is an ordered field and a € F is such that a > 0, so is a-¹.