Question

In: Advanced Math

let d be a positive integer. Prove that Q[sqrt d] = {a + b sqrt d|...

let d be a positive integer. Prove that Q[sqrt d] = {a + b sqrt d| a, b is in Q} is a field. provide explanations.

Solutions

Expert Solution


Related Solutions

8.Let a and b be integers and d a positive integer. (a) Prove that if d...
8.Let a and b be integers and d a positive integer. (a) Prove that if d divides a and d divides b, then d divides both a + b and a − b. (b) Is the converse of the above true? If so, prove it. If not, give a specific example of a, b, d showing that the converse is false. 9. Let a, b, c, m, n be integers. Prove that if a divides each of b and c,...
Let a and b be positive integers, and let d be their greatest common divisor. Prove...
Let a and b be positive integers, and let d be their greatest common divisor. Prove that there are infinitely many integers x and y such that ax+by = d. Next, given one particular solution x0 and y0 of this equation, show how to find all the solutions.
(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b)...
(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b) Prove that if m is composite, then Z/mZ is not a field. (4) Let m be an odd positive integer. Prove that every integer is congruent modulo m to exactly one element in the set of even integers {0, 2, 4, 6, , . . . , 2m− 2}
Let t be a positive integer. Prove that, if there exists a Steiner triple system of...
Let t be a positive integer. Prove that, if there exists a Steiner triple system of index 1 having v varieties, then there exists a Steiner triple system having v^t varieties
Let n be a positive integer. Prove that if n is composite, then n has a...
Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )
Let G be an abelian group and n a fixed positive integer. Prove that the following...
Let G be an abelian group and n a fixed positive integer. Prove that the following sets are subgroups of G. (a) P(G, n) = {gn | g ∈ G}. (b) T(G, n) = {g ∈ G | gn = 1}. (c) Compute P(G, 2) and T(G, 2) if G = C8 × C2. (d) Prove that T(G, 2) is not a subgroup of G = Dn for n ≥ 3 (i.e the statement above is false when G is...
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))
Let A = {a+b*sqrt14: a,b∈Z}. Prove that A ∩ Q = Z. Explain is set A...
Let A = {a+b*sqrt14: a,b∈Z}. Prove that A ∩ Q = Z. Explain is set A countable?
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
Suppose a is a positive integer and p is a prime/ Prove that p|a if and...
Suppose a is a positive integer and p is a prime/ Prove that p|a if and only if the prime factorization of a contains p. Can someone please show a full proof to this, thank you.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT