Math 361, Spring 2015, Assignment 1

Carefully define the following terms, then give one example and one non-example of each:

1. The ring $R[x]$ (where $R$ is a commutative ring).
2. Degree of a polynomial (including the degree of the zero polynomial).
3. The field $D(x)$ (where $D$ is an integral domain).

Carefully state the following theorems (you do not need to prove them):

1. Theorem on the degree of the product of two polynomials.
2. Universal mapping property of the polynomial ring $R[x]$.

Solve the following problems:

1. Section 22, problems 7, 9, and 11.

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Questions:

Is there an equivalent theorem in the book to the Universal Mapping Property in the Theorems section?

Theorem 22.4 is close, but isn't quite what I had in mind. Here is the statement: If $R$ is a commutative unital ring, $S$ is any commutative ring, $\phi:R\rightarrow S$ is any ring homomorphism, and $a$ is any element of $S$, then there exists a unique ring homomorphism $\widehat{\phi}:R[x]\rightarrow S$ extending $\phi$ (this means taking the constant polynomial $r$ to $\phi(r)$) and sending $x$ to $a$. The map $\widehat{\phi}$ necessarily has the formula $\widehat{\phi}(r_0 + r_1x + \dots + r_nx^n) = \phi(r_0) + \phi(r_1) a + \dots + \phi(r_n) a^n$, and for this reason it is referred to as a "generalized evaluation homomorphism." - Steven.Jackson (talk) 10:03, 5 February 2015 (EST)