2. Show that Q√3 = {a+b√3|a, b = Q} is a field by verifying the field axioms one by one.
2. Show that Q√3 = {a+b√3|a, b = Q} is a field by verifying the field axioms one by one.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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[Classical Geometry] How do you solve #2? 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%2F42088e93-5ac0-402b-b28d-528fa1d95fc5%2Fdcvnksn_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:Q2.
"How do I verify step by step that something is a field?"
Here are my suggestions:
Let's recall the definition first:
A FIELD is a set F equipped with two functions :
(a,b)→a+b
"+": F × F→F,
".": F × F→F, (a,b) → a. b.
satisfying the following axioms:
*************
• F1: Vx, y ≤ F,x+y=y+x.
•
F2: Vx, y, z € F, (x+y)+z=x+(y+z).
• F3: There is some element 0 € F, called the additive identity, s.t. Vx € F, x + 0 = x.
• F4: Vx € F, there is an element - x = F, called the additive inverse of x, s.t. x + (−x) = 0.
• F5: Vx, y ≤ F,x· y = y ·x.
.
.
• F6: Vx, y, z = F, (x · y) · z = x · (y • z).
-1 – 1.
-1
=
• F7: There is some element 1 € F, called the multiplicative identity, s.t. Vx € F, x·1=x.
• F8: Vx € F, x = 0, there is an element ௹ € F, called the multiplicative inverse of x, s.t. x · x¯
• F9: Vx, y, z = F, (x+y) · z = x·z+y•z.
3. Now for those 9 axioms. In most of times:
• F1 and F5: Compute directly, just labour work.
• F2 and F6: Compute directly, just labour work.
• F9: Compute directly, just labour work.
• F3 and F7: (TECHNICAL PART) This has to be done in 2 steps:
o First, what is the 0 (resp. 1) element? You need to find the 0 (resp. 1) element.
• Second, say you already got the 0 (resp. 1). Somehow is this 0 (resp. 1) necessarily in F? You need to check it.
• F4 and F8: (TECHNICAL PART) This has to be done in 2 steps:
o First, given an x E F, what is the additive (resp. multiplicative) element? You need to find this element.
• Second, say you already got the additive (resp. multiplicative) inverse for x. Somehow is this inverse necessarily in F? You need to check it.
Most of the time (to make your life easy), we will only discuss a set FC R, inheriting the two operations from those of R. So the 0 element and 1 element are just those in R as well
(why?). Hence in this case, F1&F5, F2&F6, F9, and the first part of F3 & F7 are free to you. However, you do need to check that 0, 1 are in F (you only know they are in R!), similarly for
the inverses -- after all, having an inverse in R does NOT mean having an inverse in F.
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