f09_mthsc852_hw11

# f09_mthsc852_hw11 - ms 2(a Show that N xy = N x N y(b Show...

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MTHSC 851/852 (Abstract Algebra) Dr. Matthew Macauley HW 11 Due Friday, August 28, 2009 (1) Let R = { a + b - 5: a, b Z } ⊆ C . (a) Show that R is an integral domain with 1. (b) Show that U ( R ) = 1 } . (c) Show that 3 is irreducible in R . (d) Show that a = 2 + - 5 and b = 2 - - 5 are both irreducible in R . (e) Conclude that 3 - 2 + - 5 and 3 - 2 - - 5 in R . (f) Conclude that 3 is irreducible but not prime in R , thus R is not a PID. (2) Let m N be square-free. (a) Show that Q [ m ] = { r + s m : r, s Q } , and that Q [ m ] is a ﬁeld. It is thus its own ﬁeld of fractions, which we will denote by Q ( m ). (b) Show that R m is an integral domain with 1. (c) Show that Q ( m ) is the ﬁeld of fractions for R m . (d) Show that R m is the set of all those r + s n Q ( m ) that are roots of a monic quadratic polynomial x 2 + cx + d Z [ x ]. [This is the reason for the variation in the deﬁnition of R m when m 1 (mod 4).] (3) For any x = r + s m Q ( m ), deﬁne the norm of x to be N ( x ) = r 2 -
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Unformatted text preview: ms 2 . (a) Show that N ( xy ) = N ( x ) N ( y ). (b) Show that N ( x ) ∈ Z if x ∈ R m . (c) Show that u ∈ U ( R m ) if and only if N ( u ) = ± 1. (d) Use (c) to show that U ( R-1 ) = {± 1 , ± i } , U ( R-3 ) = {± 1 , ± (1 ± √-3) / 2 } , and U ( R m ) = {± 1 } for all other negative square-free m in Z . (4) Let a and b be nonzero elements of a Euclidean domain such that a | b and d ( a ) = d ( b ). Show that a and b are associates. (5) Prove that if m =-3,-7, or-11, then R m is Euclidean with d ( r ) = | N ( r ) | for all nonzero r ∈ R m . [Hint: Mimic the proof of Proposition 3.7 from class, but choose d ∈ Z nearest to 2 t and then c ∈ Z so that c is as near to 2 s as possible with c ≡ d (mod ) , then set q = ( c + d √ m ) / 2.]...
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