Posted: November 19th, 2015

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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