1.12. AN APPLICATION TO CRYPTOGRAPHY 79 1.11.3. A Laurent polynomial is a ”polynomial” in which negative as well as positive powers of the variable x are allowed, for example, p.x/ D 7x ± 3 C 4x ± 2 C 4 C 2x . Show that the set of Laurent polynomials with coefﬁcients in a ﬁeld K forms a ring with identity. This ring is denoted by KŒx;x ± 1 Ł . (If you prefer, you may take K D R .) What are the units? 1.11.4. A trigonometric polynomial is a ﬁnite linear combination of the functions t 7! e int , where n is an integer; for example, f.t/ D 3e ± i2t C 4e it C i p 3e i7t . Show that the set of trigonometric polynomials is a sub-ring of the ring of continuous complex–valued functions on R . Show that the ring of trigonometric polynomials is isomorphic to the ring of Laurent polynomials with complex coefﬁcients. 1.11.5. Show that the set of polynomials with real coefﬁcients in three variables, R Œx;y;zŁ is a ring with identity. What are the units? 1.11.6.
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.