Why Read This Book? - Index of

9.15 Quotient Rings 341
Because the polynomial ring over a field is Euclidean, it makes for some particularly
important quotient rings. Let’s spend some time studying the quotient ring
created by “modding” out the ideal generated by f = 2t 3 − t + 5 from Q[t]. The
ties back to the integers and the integers modulo n are uncanny. To be concrete,
we will let n = 6 and draw parallels between the relationship of the integers to the
integers modulo 6 and the relationship of Q[t] to Q[t]/(f ).
Since the integers are a Euclidean domain, every integer can be written as
6q + r for integers q and r and where 0 ≤ r ≤ 5. Consequently, every integer is
equivalent mod 6 to some element of {0, 1, 2, 3, 4, 5}, so that every element of Z6 is
a coset that can be addressed by a unique representative element in {0, 1, 2, 3, 4, 5}.
To perform addition and multiplication in Z6, a purist would write something like
or
[(6) + 4]+[(6) + 3] =(6) + 7 = (6) + 1 (9.72)
[(6) + 5]×[(6) + 4] =(6) + 20 = (6) + 2 (9.73)
where the first step in these two calculations is an application of the definitions of
addition and multiplication in Z6 from Theorem 9.15.2 and the second step is an
application of equivalence mod 6 to simplify the calculation to a standard form
with representative element from {0, 1, 2, 3, 4, 5}. As long as we realize that this
is what we’re doing, we can write these calculations as
4 + 3 =6 7 =6 1 and 5 × 4 =6 20 =6 2 (9.74)
This form has the imagery of performing addition and multiplication in the integers,
with the stipulation that any sum or product that gets kicked out of bounds, that is,
out of {0, 1, 2, 3, 4, 5}, is hauled back into {0, 1, 2, 3, 4, 5} by subtracting a multiple
of 6. All this helps us see how we may view elements of Z6 and how they combine
by addition and multiplication to produce other elements of Z6.
For our specific polynomial f = 2t 3 − t + 5, the way to visualize elements
of Q[t]/(f ) in terms of polynomials in Q[t] is strikingly similar. Since Q[t] is a
Euclidean domain, any polynomial can be written uniquely as fq + r for polynomials
q and r, where either r = 0 or deg r