1. Let Z[√2] = {s+t√2 | s,t € Z}. Show that Z[√2] is a subring of R and that 1 + √2 is a unit in Z[√2].

Q: Consider Z[x] = {a,x" + a₁₁x¹+...a₁x+ao: a; € Z,i = 0,1,2,...,n), the set of all polynomials in…

A: Inverse elements are 1 and -1

Q: Prove that S = subring of R[x]. {p(x) = R[x] | p(2) = 0 or p(3) = 0} is not

A:

Q: You are buying a car. Would you prefer a higher rate for a lower term or a lower rate for a longel…

A: Given that You are buying a car. Would you prefer a higher rate for a lower term or a lower rate for…

Q: 1. Factorize each of the following polynomials over Z5 [x]. a) f(x) = x³ + 3x² + 3x + 2 b) f(x) = x¹…

A: The important thing to observe is that we have to check polynomials modulo 5 while factoring over…

Q: 6. Let f be entire and suppose that Re(f(z)) + Im(f(z)) ≥ 0. Show that f(z) must be a constant…

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Q: a. Give the smallest extension of Q that contains all the zeros of x4 - 2x2 – 15.

A: We shall answer first question only as you have asked diff question without mentioning. For others…

Q: Let F be a finite extension of Q. Prove that there are only a finitenumber of roots of unity in F.

A: Given: F is a finite extension of ℚ.To prove: There are only a finite number of roots of unity in F.

Q: B₁ = {(a, b): a, b are realt B₂B RIB is finite} I, is sopology generated by J₂ is Sopology generated…

A:

Q: assume that Pr[A]=0.25Pr[A]=0.25, Pr[B]=0.35Pr[B]=0.35, and Pr[A∩B]=0.1Pr[A∩B]=0.1. 1)Pr[A|B] =…

A: Given: P(A) =0.25 P(B)=0.35 P(A∩B) =0.1

Q: . Show that ||(1, 2)||1₁ = |x1|+|x₂| is a norm on R², but that it does not satisfy the parallelogram…

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Q: Show that, for all constructible numbers y € R, x³ – 2 is irreducible over • Q(r).

A:

Q: Let a be a complex zero of x2 + x + 1 over Q. Prove thatQ(√ a)= Q(a).

A: Given equation is  x2+x+1 Let a be the complex roots over ℚ.

Q: Manuben

A: The objective of this question is to prove that the set Z[w] = {a+bw | a, b are integers} is a…

Q: d. y= x² + y x + cos y

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Q: Let A = {x ∈ R| x≠0}, and then define f: A->R by:           f(x) = 6x+10/2x         is f…

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Q: Let x = (13, −20, 15), v1 = (1, −2, 3), and v2 = (−1, 1, 1). Then the set {v1, v2} is orthogonal. It…

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Q: Find the norm of f(x) = √√1-x² on the interval [- 1,1]. 2√3 3 O O O W|N H

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Q: Verify that the operation {x, y} = x1y1 − x1y2 − x2y1 + 3x2y2 where x = (x1, x2) and y = (y1, y2) is…

A: Given : The operation on ℝ2 is defined as  x, y = x1y1 - x1y2 - x2y1 + 3x2y2 ,  where x = (x1 , x2)…

Q: Identify R+1 with the set Pn of polynomials of degree ≤n, via the corre- spondence (any ao) → ant +…

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Q: is a maximal ideal of Q[x).

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Q: Prove that S = {p(x) = R[x] | p(2) = 0 and p(3) = 0} is a subring of R[x].

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Q: hat subring R[x², x°] not isomorphic to R[x].

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Q: Q[V5]is Principal Ideal Domain. True O False

A: We know that every field is principal ideal domain

Q: 5. а) Factor the polynomial f (x) = 3x*+2a³ +2x² +2x– 1 into irreducible polynomials %3D in C[r],…

A: "Since you have asked multiple questions, we will solve the first question  for you. If you want any…

Q: Find the inverse of 1 + √2 in Z[√2]. What is the multiplicativeorder of 1 + √2?

A: Here ℤ2=a+b2/a, b∈ℤ We need to find multiplicative inverse of 1+2 in  ℤ2. An element b∈R is said to…

Q: (a) Demonstrate that a polynomial is completely determined by its coefficients. That is, if n and m…

A: (d) Given that m1,m2,b1,b2 are real no. we need to express lm1,b1.lm2,b2 in quadratic form. We'll…

Q: The polynomial f = x³ + x + 1 is irreducible over Z3. Let u be a root of f in some extension Z3(u)…

A: Irreducible polynomial

Q: Show that u = √3+ √2i is algebraic over Q and type its minimal polynomial p(x) = irr(u, Q) below.

A: This question is about group theory.

Q: Show that u = √3+ √2i is algebraic over Q and type its minimal polynomial p(x) = irr(u, Q) below

A:

Q: 1 0 1 ~EHEMMEHO V 4 2 0 Let v = 1 0 4 O C.S O d. {v, - and S= { v₁ O a. (v. v₂} 1 2 9 O b. {V₁, V 3…

A: Since you have posted multiple questions, we will provide the solution only to the first question as…

Q: rove or Disprove that the polynomial x⁴ − 2 ∈ Q[x] is solvable by radical over Q

A: Given, x4-2∈Qx. The equation x4-2 can be reduced to a cubic equation and as it is known that any…

Q: (1 x x²). Let F be the 2: Consider the polynomial space P₂ with the standard base V = corresponding…

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Q: 9. Let D be an integral domain such that (x + 1)² = x² + 1 in D[x]. Char D = а) 0 b) 1 c) 2 d) 3 O…

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Q: Prove that S = subring of R[x]. {p(x) = R[x] | p(2) = 0 or p(3) = 0} is not

A:

Q: Which of the following is indeterminate at x = 1? x² + 1 x2 – 1 x2 – 1 Vx +3 - 2 x2 +1 Vx + 3 – 2 x…

A: We plug x=1 in each expression