Polynomials over a eld
The previous exercises showed the close relationship of nite elds to polynomials. In this part, we shall
focus on polynomials. Armed with a few key results, we will then return to the question of classifying
all nite elds. For this
2. To satisfy closure, an element + 1 must exist. Show that if + 1 cfw_0, 1, , then the resulting
structure is not a eld. (Each of the three cases yields a contradiction.)
Thus + 1 must be a fourth element of the eld.
3. The value of (a.k.a. 2 ) must also
Finite Fields:
An introduction through exercises
Jonathan Buss
Spring 2014
A typical course in abstract algebra starts with groups, and then moves on to rings, vector spaces,
elds, etc. This sequence may give the impression that elds form an advanced and
Finite Fields:
An introduction through exercises
Jonathan Buss
Spring 2014
A typical course in abstract algebra starts with groups, and then moves on to rings, vector spaces,
elds, etc. This sequence may give the impression that elds form an advanced and
Some basic nite elds
The most familiar elds are innite: the rationals, the reals, etc. But nite elds (that is, elds with
only nitely many elements) also exist, as we now explore. For this section, let F denote an arbitrary
nite eld.
F must contain 0 and 1
CS764 Computational Complexity Spring 2014
Basic problems (required): submit solutions by Wednesday, May 28.
Advanced problems (optional): submit solutions by end of lecture period.
Basic problems
These problem can be solved using the results and techniqu