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: Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (х, у)…

A:

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: Prove that Zo is not isomorphic to Z, X. 10 12

A:

Q: If a and b belong to Z[√d], where d is not divisible by the square ofa prime and ab is a unit, prove…

A: Let d be an integer that is not divisible by the square of a prime. Let a and b belong to Zd and ab…

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

A:

Q: Suppose f is entire and there exists M > 0 such that |ƒ(z)| ≥ M for all z € C. Prove that f is…

A: We have been given that f is entire and there exists M>0 such that fz≥M for all z∈ℂ. To Prove: f…

Q: ZW and ZX are supplementary and ZW and ZY are supplementary. Construct a proof that ZX = ZY.

A: <W  and < X  are supplementary angle  the sum of supplementary angles = 180 degree <W  and…

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: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

A:

Q: Let a, b, and c be distinct real numbers. Show that…

A: Consider the equation: We have to check whether it is a zero polynomial, symmetric and bi-linear or…

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: Let D be an integral domain. (i) If aD is maximal,then show that 'a' is irreducible (ii) If D is a…

A:

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

A:

Q: 9. Show that if a is a root of X²-bX+c with b, c integers and 6² - 4c < 0, then there are only…

A: Given that α is a root of X2-bx+c, with b,c integers and b2-4c&lt;0. Since b2-4c&lt;0, therefore…

Q: Show that |Re(z)| + |Im(z)| ≤ √ 2|z| for any z ∈ C

A: We will use the basic knowledge of the complex number system to answer this question.

Q: Let a be a non-zero element of Z43. w| 42. Show that [3] is a primitive root of Z43 and that [2] is…

A:

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

A:

Q: Let R be a principal ideal domain. For any two distinct nonzero elements ƒ and g with no common…

A:

Q: 3. Show that Z[√2i] is a Euclidean Domain with the function N(z) = |zz|.

A: Given function Nz=zz¯ We need to show that ℤ2i is a Euclidean Domain with the function. A Euclidean…

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

A:

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…

A:

Q: Let V be the set of all pairs (x, y) of real numbers together with the following operations: (x₁,…

A: V be the set of all pairs (x,y) of real numbers  together with the following operations ;…

Q: Find a zero-divisor in Z,[i] = {a + bi | a, b E Z3}.

A:

Q: Determine if the following polynomials are- irreducible in

A: In the given question, the concept of Irreducible Polynomial is applied. Irreducible Polynomial An…

Q: Q[V2 ]is Principal Ideal Domain..

A: Q[√2] is Principal Ideal Domain. True.

Q: P10.2. Let w := }(−1 + √√3i) € C be one of the complex roots of the polynomial x² + x + 1 E in C[x].…

A: Advance maths

Q: Claim that = (0.0.0) 2 a) show √ E OF √₂, √₂. b) Show OF J₁ () show J₁ s={√₁₁ √√₂, √√₂} S= √√₂₁ are…

A:

Q: a) Factor a +x³ + x into irreducible polynomials in Z2[r];

A: i) Linear factors are irreducible. ii) Just substitute values in Zp and see the expression becomes…

Q: Let a and b belong to Q with b ≠ 0. Show that Q(√a) = Q(√b) ifand only if there exists some c ∈ Q…

A: Given: a and b belong to Q with b ≠ 0. To Prove: Qa=Qb if and only if there exists some c∈Q such…

Q: Show that I+J is an ideal of L where I+J ={x+y:x∈I, y∈J}.

A: Given that and  are two ideals of a ring Ideal Test:A nonempty subset A of a ring is an ideal of…