College Algebra Exam Review 68

# College Algebra Exam Review 68 - 78 1 ALGEBRAIC THEMES The...

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Unformatted text preview: 78 1. ALGEBRAIC THEMES The ﬁrst assertion is this: If x Á y .mod ab/, then x Á y .mod a/ and x Á y .mod b/. But this is actually evident, as it just says that if x y is divisible by ab , then it is divisible by both a and b . The second assertion is similar but slightly more subtle. We have to show if Œxab ¤ Œyab , then .Œxa ; Œxb / ¤ .Œya ; Œyb /; that is, Œxa ¤ Œya or Œxb ¤ Œyb . Equivalently, if x y is not divisible by ab , then x y is not divisible by a or x y is not divisible by b . The (equally valid) contrapositive statement is: If x y is divisible by both a and b , then it is divisible by ab . This was shown in Exercise 1.6.11. Now, since both Zab and Za ˚ Zb have ab elements, a one-to-one map between them must also be onto. This means that, given integers ˛ and ˇ , there exists an integer x such that .Œxa ; Œxb / D .Œ˛a ; Œˇb /; or Œxa D Œ˛a and Œxb D Œˇb . But this means that x Á ˛ .mod a/ and x Á ˇ .mod b/. I We didn’t need it here (in order to prove the Chinese remainder theorem), but the map from Zab to Za ˚ Zb deﬁned by Œxab 7! .Œxa ; Œxb / is a ring isomorphism (See Exercise 1.11.10). A ﬁeld is a special sort of ring: Deﬁnition 1.11.7. A ﬁeld is a commutative ring with multiplicative identity element 1 ¤ 0 (in which every nonzero element is a unit. Example 1.11.8. (a) R, Q, and C are ﬁelds. (b) Z is not a ﬁeld. RŒx is not a ﬁeld. (c) If p is a prime, then Zp is a ﬁeld. This follows at once from Proposition 1.9.9. Exercises 1.11 1.11.1. Show that the only units in the ring of integers are ˙1. 1.11.2. Let K be any ﬁeld. (If you prefer, you may take K D R.) Show that the set KŒx of polynomials with coefﬁcients in K is a commutative ring with the usual addition and multiplication of polynomials. Show that the constant polynomial 1 is the multiplicative identity, and the only units are the constant polynomials. ...
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