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College Algebra Exam Review 69

College Algebra Exam Review 69 - 1.12 AN APPLICATION TO...

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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 KOEx; 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 OEx; y; zŁ is a ring with identity. What are the units?
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