3. Let R be the subring (a+b√10 | a,b e Z) of the field of real numbers. (a) The map N:R-Z given by a+b√10(a+b√10)(a - bic =a² - 10b² is such that N(uv) = N(u)N(v) for all u,v e R and N(u) = 0 if ar only if u = 0. ✓(b) u is a unit in R if and only if N(u) ±1.

3. Let R be the subring (a+b√10 | a,b e Z) of the field of real numbers. (a) The map N:R-Z given by a+b√10(a+b√10)(a - bic =a² - 10b² is such that N(uv) = N(u)N(v) for all u,v e R and N(u) = 0 if ar only if u = 0. ✓(b) u is a unit in R if and only if N(u) ±1.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.3: The Field Of Quotients Of An Integral Domain

Problem 10E: Since this section presents a method for constructing a field of quotients for an arbitrary integral...

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3. Let R be the subring (a+b√10 | a,b e Z) of the field of real numbers.
(a) The map N:R-Z given by a+b√10+(a+b√10)(a - b√ic
a²- 10b² is such that N(uv) = N(u)N(v) for all u,ve R and N(u) = 0 if ar
only if u = 0.
=
(b) u is a unit in R if and only if N(u) = ±1.
(c) 2, 3, 4+ √10 and 4 - √10 are irreducible elements of R.
(d) 2, 3, 4 + √10 and 4-
-
-(4+√10)(4-√10).j
=
10 are not prime elements of R. (Hint: 3-2 =

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(c) 2, 3, 4+ √10 and 4 - √10 are irreducible elements of R.
(d) 2, 3, 4 + √10 and 4-√10 are not prime elements of R. [Hint: 3-2
= (4+ √10)(4-√10).j

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