Let R be the subring fatby 10 | a, bE Z, } of the field of real numbers. "( 2) The map N: R D Z given by at blot ( at by 10) ( a- bvIO ) = Q2- 10b is such that N(ur ) = Nfu) (v ) for all LIVER and Ncu) = 0 if and only if 4= 0. ( ) u is a unit in R if and only if N ( u ) = +1. (c ) 2, 3, 4 + 1/10 and 4- V10 are irreducible elements of R d = dad (2) ( d) 2. 3, 4 + 10 and 4- 710 are not prime elements of R. Link 3.2 = 6 = 4 +170 ) ( 4 170) HINT: for part ( ) : you may ine ( a = 161 10, asb E Z Did= b= 0. 40 for port ( b ) : you may une N ( 4) N ( V ) = 1 NCY, NCV ) E Z N (U) = + 1 and at by To ) ( a - bvIO ) = 72 106? for part ( ) : you may une the following : N ( y ) N() = 40 => either Nu) = + 1 or f2 or +4 . If N ( u ) = 12 the a2 - ( +2 ) = 1062 82 = +2 ( mod 5) . Write a= 59to where ours 4 . Then 12 = f 2 ( mod 5 ) , which is a contradiction. For part () : you may use ( 41VI0) ( 4- 110) = 6 = 2.3 and 2,3 + ( 4+110)