Show that the set of all real numbers of the form a+b sqrt(2)+c sqrt(3)+d sqrt(6),where a,b,c,d...

Show that the set of all real numbers of the form a+b sqrt(2)+c sqrt(3)+d sqrt(6),where a,b,c,d ∈Q, forms a subfield of R

Expert Solution

Colby Messinger answered 2 years ago

let Q(sqrt(2),sqrt(3)) be the field generated by elements of the form a+bsqrt(2)+csqrt(3)+dsqrt(6) where a,b,c,d are in...

let Q(sqrt(2),sqrt(3)) be the field generated by elements of the form a+bsqrt(2)+csqrt(3)+dsqrt(6) where a,b,c,d are in Q(sqrt(2),sqrt(3)) is a vector space of dimension 4 over A. find a basis for Q(sqrt(2),sqrt(3))

Prove that the set R ={ a+ b√2+c√3+d√6 , a,b,c,d belongs to Q } is a...

Prove that the set R ={ a+ b√2+c√3+d√6 , a,b,c,d belongs to Q } is a field

prove that where a,b,c,d,e are real numbers with (a) not equal to zero. if this linear...

prove that where a,b,c,d,e are real numbers with (a) not equal to zero. if this linear equation ax+by=c has the same solution set as this one : ax+dy=e, then they are the same equation.

Consider the parabolas y=x^2 and y=a(x-b)^2+c, where a,b,c are all real numbers (a) Derive an equation...

Consider the parabolas y=x^2 and y=a(x-b)^2+c, where a,b,c are all real numbers (a) Derive an equation for a line tangent to both of these parabolas (show all steps with a proof, assuming that such a line exists) (b) Assume that the doubly-tangent line has an equation y+Ax+B. Find an example of values of a,b,c (other than the ones given here) such that A,B ∈ Z

Consider the parabolas y=x^2 and y=a(x-b)^2+c, where a,b,c are all real numbers (a) Derive an equation...

Consider the parabolas y=x^2 and y=a(x-b)^2+c, where a,b,c are all real numbers (a) Derive an equation for a line tangent to both of these parabolas (show all steps, assuming that such a line exists) (b) Assume that the doubly-tangent line has an equation y+Ax+B. Find an example of values of a,b,c (other than the ones given here or in class) such that A,B ∈ Z (c) Consider the case a = 1. Does your algebra accurately reflect what we can...

On the set S of all real numbers, define a relation R = {(a, b):a ≤ b}. Show that R is transitive.

Find the value of a : b : c : d, if a : b = 2 : 3, b : c = 4 : 5 and c : d = 6 : 7.

1. Evaluate: (a+b)/(c-d) + 9/(a+d) when a=5, b=3, c=8, d=4 a. 6 b. 3 c. 15/2...

1. Evaluate: (a+b)/(c-d) + 9/(a+d) when a=5, b=3, c=8, d=4 a. 6 b. 3 c. 15/2 d. 17/13 2. Solve for x: 5(x+3) = 35 a. 2 b. 7 c. 4 d. -4 3. Acid rain occurs primarily as a result of a. operating a nuclear power plant b. burning coal or oil containing sulfur c. by-products created by operating an oil refinery d. the use of Freon and other refrigerants 4. The "ozone holes" at the polar region arise...

2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff...

2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements. 3. Define a relation R on integers as follows: mRn iff m + n is even. Is R a partial order? Why or why not? If R is...

5. (a) Prove that the set of all real numbers R is uncountable. (b) What is...

5. (a) Prove that the set of all real numbers R is uncountable. (b) What is the length of the Cantor set? Verify your answer.

Question

Show that the set of all real numbers of the form a+b sqrt(2)+c sqrt(3)+d sqrt(6),where a,b,c,d...

Solutions

Expert Solution

Related Solutions

let Q(sqrt(2),sqrt(3)) be the field generated by elements of the form a+bsqrt(2)+csqrt(3)+dsqrt(6) where a,b,c,d are in...

Prove that the set R ={ a+ b√2+c√3+d√6 , a,b,c,d belongs to Q } is a...

prove that where a,b,c,d,e are real numbers with (a) not equal to zero. if this linear...

Consider the parabolas y=x^2 and y=a(x-b)^2+c, where a,b,c are all real numbers (a) Derive an equation...

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Find the value of a : b : c : d, if a : b = 2 : 3, b : c = 4 : 5 and c : d = 6 : 7.

1. Evaluate: (a+b)/(c-d) + 9/(a+d) when a=5, b=3, c=8, d=4 a. 6 b. 3 c. 15/2...

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