Let R = Z[√2] = {a+b√2: a,b € Z} and F = Q[√2] = {a+b√2: a,b € Q}. Then R is a ring, F is a field, and RCFCR. Define the norm on F by setting v(a+b√2)= |a²-26²1. Note that for a = a +b√2, v(a)= |aa| where aa = a - b√2 (a) Show that for any a, ß € F, aß = aß and use this to show that v(aß) = v(a)v(B)

Q: (15 points) Prove the ring Z[√−2] = {a+b√−2 | a,b ≤ Z} is a Euclidean domain with norm 8(a+b√2) = a²…

A:

Q: a Let Z[√2] = {a+b√√2 | a, b € Z} and let H = { b Show that Z[√2] and H are isomorphic as rings. 2b…

A:

Q: Q2) Let (M₂ (R), +..) be a ring. Prove H = {(a) la, b, c €. = {(ab)la, b, c € R}is a subring of (M₂…

A: Subring Test : A non empty subset S of a ring R is a subring if S is closed under subtraction and…

Q: Let R be an integral domain. a) Show that if R is normal, then so is S-1R for any multiplicatively…

A: Given: S-1R=rs: r∈ℝ, s∈ℝ Let a,b∈S-1ℝ. a∈S-1R and b∈S-1R thena=r1s1 & b=r2s2 for some r1,r2∈R…

Q: Let x, y ∈ F, where F is an ordered field. Suppose 0 < x < y. Show that x2 < y2.

A: multiplying by x and y in x&lt;y respectively and then comparing both result.....

Q: (d) Show that the division algorithm holds in Z[V2]. Hint: Let a, 3 e Z[v2] with B+ 0. Write a/3 =r+…

A:

Q: Let (R, +,.) be any ring. Define R × Z = {(r,n):r E R ,n E Z} and %3D (r, n) + (s, m) = (r + s,m +…

A: In the Structure T=ℝ×ℤ additional multiplication is defined as…

Q: やthe ring KLx,り,z3 wherc K is a field. Prove (x2 - (y1)) that is a K[x, り,そI . Prime ideal of

A: Given:- Ring  K[x ,y ,z]   where k is a field. To Prove:-  [x z-(y2+1)]

Q: a) The ring R[x, y]/(x + 1) is a field. b) The ring Z[x]/(7) is a principal ideal domain

A: a) The ring R[x, y]/(x + 1) is not a field. To determine if it is a field, we need to check if every…

Q: For the ring Z[Vd] = {a + bVd | a, b E Z}, where d + 1 and d is not divisible by the square of a…

A:

Q: Let F be a field and aeF be such that [F (a): F]=5. Show that F(a)= F(x³).

A:

Q: Qz: Let Z be the set of integer numbers and *, o defined as follows Va, be Z, then a *b = a +b+1 a…

A: See the attachment

Q: Show that in the factor ring $Z[x]/\langle 2x + 1 \rangle$,the element $x + \langle 2x + 1 \rangle$…

A:

Q: Show that the points of intersection of a circle in the plane of a fieldF and a line in the plane of…

A: We have to show that the points of intersection of a circle in the plane of a field F and a line in…

Q: Let R be a ring with unity. Show that (a) = { E xay : x, y e R }. finite

A:

Q: = Let I andJ be two ideals of a commutative ring R. Show that: S= {r€R: ri E J Vi EI } is an ideal…

A: Ideal: A subset of a ring is called ideal of if  1. 2. Given: are ideals in a ring and . To…

Q: a. Show that the field Q(v2, V3) = {a + bv2 + cv3 + dvZV3: a, b, c, d E Q} is a finite extension of…

A: a. The field extension Q(√2, √3), obtained by adjoining √2 and √3 to the field Q of rational…

Q: Rp { -b a | a, b = Zp}

A: It is given that,  Rp = ab-ba : a, b ∈ ℤp We have to prove that,  Rp is a commutative ring with unit…

Q: Let R be a ring with unity. Show that (a) = { £ xay: x, y e R }. finite

A:

Q: 2. Let K be a field and let ƒ € K be a nonsquare (i.e. there is no element a € K such that f = a²).…

A:

Q: Let Q be the field of rational numbers and let F √2, √7, √14 are irrational. = Q(√2, √7). You may…

A:

Q: Let a ER which expressions are correct? and give the justification why they are part of the Borel…

A:

Q: (1) Let a be a fixed element in the commutative ring R, and let A- (xe R: xa = 0). Prove that A is…

A:

Q: Prove that in a Euclidean ring R, (a, b) can be found as follows : b = q0 a + r¡, where d (r¡)<d (a)…

A:

Q: Let S = {a+b√2| a, b ≤ Q}. Is (S, +, ×) a ring? Prove your result.

A:

Q: Let V denote the space of polynomials in x of degree ≤ 3 over a field K. Let v1 =1+x+x2 +x3, v2…

A: Introduction  We have given  the space of polynomials in x of degree ≤ 3 over a field K. Let v1…

Q: a. Show that the field Q(vZ. v3) = (a+byZ +cv3+ dvZ3: a, b,c, d e Q) is a finite %3D extension of Q.…

A:

Q: Consider Z[√2] = {a + b√2 | a,b € Z}. Determine whether (Z[√2], +, •) is a ring.

A:

Q: Q2) Let(M₂ (R), +..) be a ring. Prove H = {(a) la, b, c = R}is a subring of (M₂ (R), +,.).

A:

Q: explain if the following statement is true or false in detail: "The ideal (x + 1) is maximal in the…

A: "The ideal (x + 1) is maximal in the ring R[x, y]"

Q: (b) Note that for a E R, v(a) E N. Show that R is an Euclidean domain with respect to the norm v.…

A:

Q: Use the field norm to show: a) 1 + √2 is a unit in Z[√2] b) −1+√√-3 -1+√√-3 2 is a unit in Z [ 1+…

A:

Q: Let F be a field. Prove that Fl) E F.

A: