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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
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.
 Fall '08
 EVERAGE
 Algebra, Polynomials, Cryptography

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