# ahw9 - Math 5125 Monday October 31 Ninth Homework Solutions...

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Unformatted text preview: Math 5125 Monday, October 31 Ninth Homework Solutions 1. Let R = Z [ √- n ] where n is a square free integer greater than 3. (a) Prove that 2, √- n and 1 + √- n are irreducibles in R . (b) Prove that R is not a UFD. (c) Give an explicit ideal in R that is not principal. Define N : R → Z by N ( a + b √- n ) = ( a + b √- n )( a- b √- n ) = a 2 + nb 2 . Note that N ( αβ ) = N ( α ) N ( β ) for α , β ∈ R . Also if N ( α ) = ± 1, then we must have N ( α ) = 1, a = ± 1 and b = 0, which tells us that α = ± 1 and in particular is a unit. (a) N ( 2 ) = 4, so if 2 = αβ with α , β nonunits, then N ( α ) = 2. This is not possible because we cannot solve a 2 + nb 2 = 2 with a , b ∈ Z . Furthermore N ( √- n ) = n so if √- n = αβ , then N ( α ) N ( β ) = n which implies either N ( α ) or N ( β ) = 1 and we deduce that either α or β is a unit. Finally N ( 1 + √- n ) = 1 + n , so if 1 + √- n = αβ , then again we must have either N ( α ) or N ( β ) = 1, which shows that either α or β is a unit. We have now proven that 2, √- n and 1 +...
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ahw9 - Math 5125 Monday October 31 Ninth Homework Solutions...

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