Unformatted text preview: A . 5 . [6 pts] Prove that Z does not have an upper bound. HINT : Use LUB. 6 . [5 pts] Let α = 2 1 / 3 . Deﬁne Q [ α ] := { a + bα + cα 2 : a,b,c ∈ Q } . Prove that Q [ α ] is closed under multiplication. 7 . [6 pts] Prove that if a ∈ Z n and a and n are relatively prime then a has a multiplicative inverse in Z n . 8 . [5 pts] Find the multiplicative inverse of 19 in Z 81 . 9 . [2 + 2 + 2 = 6 pts] (a) Deﬁne the term ordered ﬁeld . (b) Give an example of an ordered ﬁeld. (c) Give an example of a ﬁeld which is not an ordered ﬁeld. 1...
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 Spring '08
 Staff
 Number Theory, Natural number, Greatest common divisor, Coprime, Fundamental theorem of arithmetic

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