Which of the following sets are rings with respect to the usual operations of addition and...

Which of the following sets are rings with respect to the usual operations of addition and multiplication? If the set is a ring, is it also a field? can you explain in detail, I really do not understand how to prove.

Q(√2,√3)={a+b√2+c√3+d√6:a,b,c,d∈Q}
R={a+b3√3:a,b∈Q}

Solutions

Expert Solution

Since for (b) R={a+b3√3:a,b€Q } is not a ring so not field

For (a) Q(√2,√3) I have proved that it is field

Since their are various ways to proof it is field.

please Rate this answer Thank you ?

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Which of the following are groups? + And · denote the usual addition and multiplication of...

Which of the following are groups? + And · denote the usual addition and multiplication of real numbers. (G, +) with G = {2^ n | n ∈ Z}, (G, ·) with G = {2 ^n | n ∈ Z}. Determine all subgroups of the following cyclic group G = {e, a, a^2, a^3, a^4, a^5}. Which of these subgroups is a normal divisor of G?

Which of the following sets are closed under addition? (i) The set of all vectors in...

Which of the following sets are closed under addition? (i) The set of all vectors in R2 of the form (a, b) where b = a2. (ii) The set of all 3 × 3 matrices that have the vector [3 -1 -1]T as an eigenvector. (iii) The set of all polynomials in P2 of the form a0 + a1 x + a2 x2 where a0 = a2. (A) (i) only (B) all of them (C) (ii) only (D) (i) and...

Which of the following sets are closed under addition? (i) The set of all vectors in...

Which of the following sets are closed under addition? (i) The set of all vectors in R2 of the form (a, b) where b = a. (ii) The set of all 3 × 3 matrices that have the vector [-1 3 -3]T as an eigenvector. (iii) The set of all polynomials in P2 of the form a0 + a1 x + a2 x2 where a0 = a22.

Which of the following sets are closed under addition? (i) The set of all vectors in...

Which of the following sets are closed under addition? (i) The set of all vectors in R2 of the form (a, b) where b = a. (ii) The set of all 2 × 2 matrices that have the vector [-2 -3]T as an eigenvector. (iii) The set of all polynomials in P2 of the form a0 + a1 x + a2 x2 where a0 = a22.

Which of the following sets of quantum numbers are not allowed? For the sets that are...

Which of the following sets of quantum numbers are not allowed? For the sets that are incorrect state what is wrong. For the sets that are correct, give the orbital designation (eg 1s, 2px). n l ml ms a) 2 1 +1 +1/2 b) 2 2 -1 -1/2 c)75-4 0 d) 4 2 -2 +1/2

Calculate the relative (sub-space) topology with respect to the usual (metric) topology in R (the set...

Calculate the relative (sub-space) topology with respect to the usual (metric) topology in R (the set of real numbers), for the following sub-sets of R: X = Z, where Z represents the set of integers Y = {0} U {1 / n | n is an integer such that n> 0} Calculate (establish who are) the closed (relative) sets for the X and Y sub-spaces defined above. Is {0} open relative to X? Is {0} open relative to Y?

Let R and S be rings. Denote the operations in R as +R and ·R and...

Let R and S be rings. Denote the operations in R as +R and ·R and the operations in S as +S and ·S (i) Prove that the cartesian product R × S is a ring, under componentwise addition and multiplication. (ii) Prove that R × S is a ring with identity if and only if R and S are both rings with identity. (iii) Prove that R × S is a commutative ring if and only if R and...

Determine if the following sets are real vector spaces with the indicated operations. (a) The set...

Determine if the following sets are real vector spaces with the indicated operations. (a) The set V of all ordered pairs (x, y) of real numbers with the addition defined by (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2 + 1) and scalar multiplication defined by α(x, y) = (αx, αy + α − 1), α ∈ R (b) The set V of all 2 × 2 real matrices X with the addition of M22 but...

Which of the following variances are most similar with respect to the manner in which they...

Which of the following variances are most similar with respect to the manner in which they are calculated? Multiple Choice Labor rate variance and labor efficiency variance. Material price variance and material quantity variance. Material price variance, material quantity variance, and total material variance. Material price variance and labor rate variance. Material price variance and labor efficiency variance.

Which of the following sets are not well defined? Explain.

Which of the following sets are not well defined? Explain. a. The set of wealthy school teachers b. The set of great books c. The set of natural numbers greater than 100 d. The set of subsets of \(\{1,2,3,4,5,6\}\) e. The set \(\{x \mid x \neq x\) and \(x \in N\}\)

Subjects