Posted: November 19th, 2015

# Ring Theory

Ring Theory
Assignment 2
1. Let F be a field, and define a function d : F[t, t-1
] -? N in the following
way: For an element f(x) = P
i?Z
ait
i
, let dmax(f) be the maximal exponent
of t that occurs in f, i.e. dmax(f) = max{i | ai 6= 0}, and let dmin(f)
be the minimal exponent of t that occurs in f, i.e. dmin(f) = min{i | ai 6=
0}; and set d(f) = dmax(f) – dmin(f).
(a) Find d(f), d(g) and d(h), where
f(t) = -7t
-6 + 4t
-3 + 5
g(t) = 2t
-3 + t
-2 – t
-1
h(t) = 12t
6
are elements of Q[t, t-1
].
(b) Show that d(fg) = d(f) + d(g) for nonzero Laurent polynomials f
and g.
(c) Show that for f ? F[t, t-1
], f 6= 0 we have that t
af lies in the subring
F[t] of F[t, t-1
] and that deg(t
af) = d(f), when a = -dmin(f).
(d) Show that for f, g ? F[t, t-1
], f, g 6= 0 we can get f = qg + r for
Laurent polynomials q and r such that either r = 0 or d(r) < d(g).
[Hint: Use the fact that F[t] is a Euclidean ring on the polynomials
t
af and t
b
g as constructed in part (c).]
(e) Show that F[t, t-1
] is a Euclidean ring.
2. In each of the following situations, find a greatest common divisor of r
and s and hence find a generator for the ideal (r) + (s).
(a) r = x
3 – 3x
2 + 3x – 2, s = x
2 – 5x + 6 in Q[x].
(b) r = 4t
-2 + t
2
, s = t
-3 + t
-1
in Q[t, t-1
].
(c) r = 1 – 5i, s = 1 + 2i in Z[i].
3. Determine if the polynomial 3x
5 + 4x
3 -20x
2 -2x+ 6 ? Q[x] is irreducible
or not.
1

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