Rings with Inquiry: An inquiry-oriented introduction to ring theory.

One of the most beautiful consequences of an abstract study of algebra is the fact that both \(\Z\) and \(F[x]\) are Euclidean domains. While they are not “the same”, we can expect them to share many of the same properties. In this section, our first goal will be to extend familiar properties from \(\Z\) to \(F[x]\text{.}\) We will also see that particular features of a polynomial (e.g., its degree, or the existence of roots) allows for additional criteria for its irreducibility to be decided.

Since both \(\Z\) and \(F[x]\) have a division algorithm, it is reasonable to expect that, similar to the integers, we can also investigate the greatest common divisor of polynomials. In fact, our method for finding the greatest common divisor of two integers extends nicely to polynomials.

One of the most useful things we can do with polynomials is evaluate them by “plugging in” elements from our coefficient set (or some superset that contains it) and performing the resulting arithmetic in an appropriate ring. We can make this completely rigorous using the language of functions: given a commutative ring \(R\) and all polynomials \(p(x) = a_0 + a_1 x + \cdots + a_n x^n \in R[x]\text{,}\) we define the function \(p_f : R\to R\) by \(p_f(r) = a_0 + a_1 r + \cdots + a_n r^n \text{.}\) However, we will not belabor this point; instead, we will generally write \(p(r)\) in place of \(p_f(r)\) and appeal to our common notions of evaluating polynomials.

Given a polynomial \(p(x)\in R[x]\text{,}\) we have frequently been interested in finding all \(r\in R\) for which \(p(r) = 0\text{.}\)

When considering polynomials with integer coefficients, any rational roots are particularly well-behaved.

Theorem 3.1.2 gave a condition to check to see if polynomials in \(\Z[x]\) had roots in \(\Q\text{.}\) However, the lack of a rational root for a polynomial \(q(x)\in \Z[x]\) is not sufficient to say that a polynomial is irreducible in \(\Z[x]\) according to Definition 2.3.4.

To simplify matters, we will focus henceforth on polynomials with coefficients in a field. The following theorem is a result that you learned in high school algebra

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www.corestandards.org/Math/Content/HSA/APR/B/2/

(and have likely used countless times since then), but as with the other familiar topics we have explored so far, it is necessary to formalize prior to continuing.

Note that while \(F[x]\) is a ring, and we already have a definition of an irreducible element of a ring, we will find it useful to have a ready definition of irreducible in the context of polynomials with coefficients in a field. It is to that task that we now turn.

The preceding theorems allow us to explore the (ir)reducibility of polynomials of small degree with coefficients in any field.

As the following theorem illustrates, in \(F[x]\text{,}\) all irreducibles are primes.

We next state the Fundamental Theorem of Algebra. Despite its name, its proof relies on analytic properties of the real numbers; there is no purely algebraic proof. Moreover, it is not essential for the work we do in following sections, but given its close relationship to the question of factorization, we include it here for completeness.

We conclude with one consequence of the Fundamental Theorem of Algebra.

Thus, the multiplicative structure of \(\C[x]\) is straightforward: everything can be factored as a product of linear polynomials. Fields of coefficients like \(\C\) for which this is true are said to be algebraically closed; not all fields satisfy this property. For instance, \(x^2 + 1\in \R[x]\) does not factor into a product of linear polynomials. Consequently, \(\R\) is not algebraically closed.

However, regardless of whether our field is algebraically closed, we have not yet determined that any \(p\in F[x]\) can be factored uniquely into a product of irreducibles, or even that such factorizations into irreducibles exist. In Section 3.2, we do just that.