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Unformatted text preview: f eld. Solution. False. (v) The subset R = Q [ i ] = { a + bi : a , b ∈ Q } is a sub f eld of C . Solution. True. (vi) The prime f eld of Q [ i ] = { a + bi : a , b ∈ Q } is Q . Solution. True. (vii) If R = Q [ √ 2 ] , then Frac ( R ) = R . Solution. False. 3.18 (i) If R is a commutative ring, de f ne the circle operation a ◦ b by a ◦ b = a + b − ab . Prove that the circle operation is associative and that 0 ◦ a = a for all a ∈ R . Solution. It is easy to see that 0 ◦ a = a for all a ∈ R : ◦ a = + a − · a = a . Let us show that ◦ is associative. ( a ◦ b ) ◦ c = ( a + b − ab ) ◦ c = a + b − ab + c − ( a + b − ab ) c = a + b + c − ab − ac − bc + ( ab ) c ;...
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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