# FACTORING POLYNOMIALS AND POWER SERIES

1) Let K be complete with respect to a non-trivial non-archimedean
absolute

value | |. Is every absolute value on K (T) which extends | | equivalent to | |_{c}
for

some
?

2) Let K be complete with respect to a non-trivial non-archimedean absolute

value | |. Let
and let
denote the subset of formal sums

with
is the set of power series over K which

“converge in a closed ball radius c”.] If
define

(a) Show that |f|_{c} = 0 if and only if f = 0.

(b) Show that K[T]_{c} is closed under addition and that

(c) Show that K[T]_{c} is closed under multiplication and that

(d) Show that K[T]_{c} is complete with respect to | |_{c}.

(e) Show that K[T] is dense in K[T]_{c}.

(f) If
and
satisfies

show that we can uniquely write

h = qf + r

where,
where
has degree less than n, and where

[Hint: use part (e).]

3) Which of the following polynomials are irreducible over Q_{5}?

4) If p is an odd prime determine modulo p^{3} the monic irreducible factors of

X^{3} + 2pX^{2} + pX + p^{2}.

5) How many roots does X^{3} + 25X^{2} + X − 9 have in Q_{p} for p = 2, 3, 5, 7?

6) Keep the notation and assumptions of question 2). If

let NP(f) denote the boundary of the smallest convex set containing the points

and (0, y) for y any sufficiently large real number .
Suppose

Let
denote the lowest line of slope
which meets NP(h). Let m

denote the largest x- coordinate of a point of intersection of
with NP(h). Why
is

m an integer? Show that we can write

h = fg

where
is a polynomial of degree m, where

and where

for all i > 0.

Deduce that there are only finitely many
with |

Also deduce that the number of zeros
with
is zero unless

NP(h) has a side of slope −d, in which case the number of such zeros is
positive,

but less than or equal the length of the x-axis below the side with slope −d.

7) Suppose that K is algebraically closed and complete with respect to a
nontrivial

non-archimedean absolute value | |. Also suppose that

converges at all elements of K. Show that f has only finitely many zeros in any

closed ball
Let
be the zeros of f in K in order of

increasing absolute value. Show that

where the limit is taken with respect to any one of the absolute values | |_{c}
with

Deduce that for any t ∈ K

8) (a) Show that

where [t] denotes the greatest integer less than or equal to a real number t.
Deduce

that

(b) If with
and if m is a positive integer recall that
we

have

Show that this can be rewritten as

where the do not depend on m and tend to zero
as

(c) Show that X^{2} + X + 3 splits as (X −α )(X −β) over
and

. Calculate

mod 27

and

mod 81

for all j.

(d) Consider the recurrence relation u(n+2) = 3u(n+1)−5u(n) with u(0) = 1

and u(1) = 2. Show that u(2m + 1) −1 mod 3 for all non- negative integers m.

Also show that

Use this to write

where and the sum converges 3-adically for all
. Calculate each

modulo 81. Show that u(2m) = 1 for at most three non-negative integers m. Find

all non-negative integers m with u(2m) = 1.

Prev | Next |