Let K be the ring that contains as a subring. Show that has no roots in K. Thus, Corollary 5.12 may be false if F is not a field. [Hint: If u were a root, then , and . Derive a contradiction.]
It is proved that has no roots in K.
Exercise 3.2.17 states that if is a unit in a ring with an identity, then will not be a zero divisor.
Assume that is a root of ; therefore, . There is , and as a result, 3 is a unit in .
Moreover, ; therefore, 3 will be a zero divisor in , and .
It is also observed that contains a root in (according to Theorem 5.11).
This contradicts the result that units in a ring with unity are not zero divisors.
Hence, it is proved has no roots in K.
In Exercises 5-8, each element of the given congruence-class ring can be written in the form [ax + b] (Why?). Determine the rules for addition and multiplication
of congruence classes. (In other words, if the product [ax + b][cx + d] is the class [rx + s], describe how to find r and s from a, b, c, d and similarly for addition.)
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