Math 360, Fall 2013, Assignment 13

"And do you do Addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?"

- Lewis Caroll, Through the Looking Glass

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

1. Field of fractions (of an integral domain).
2. Polynomial function (from a commutative ring $R$ to itself)
3. Polynomial (with coefficients in $R$)
4. Evaluation homomorphism (from $R[x]$ to $R$).
5. Zero (or root) of a polynomial.

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

1. Universal mapping property of the field of fractions.
2. Universal mapping property of $R[x]$.
3. Relationship between $R[x]$ and the ring of polynomial functions on $R$.

Solve the following problems:

1. Section 21, problems 1 and 2.
2. Section 22, problems 1, 5, 7, 11, 13, and 15.

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Field of Fractions of an Integral Domain

   Definition:

   Let \(D\) be an integral domain. Its field of fractions is the set $$ F=\{(a,b)|a,b\in D, b\neq 0\} $$

   with addition and multiplication defined by:$$ (a,b)+(c,d) = (ad+cb,bd)\\ (a,b)(c,d)=(ac,bd) $$

   Example:

   The rational numbers are the field of fractions of \(\mathbb{Z}\). (Just write \(\frac{a}{b}\) instead of \((a,b)\).)

   Non-Example:

   It is not possible to create the field of fractions of the \(n\times n\) matrices, because they are not an integral domain.

2. Polynomial Function

   Definition:

   A polynomial function in a commutative ring \(R\) is a function \(p:R\rightarrow R\) such that:$$ p(x) = \sum_{i=0}^n a_ix^i $$

   Example:

   The function:$$ p(x) = 4x^2 +3x + 2 $$

   Non-Example:

   The function:$$ p(x) = 4x^2 + 3x + 1/x $$

3. Polynomial

   Definition:

   A polynomial with coefficients in \(R\) is a sequence \((a_i)\), with \(a_i\in R\) and only finitely many non-zero \(a_i\). The set of all such polynomials is \(R[x]\).

   Example:

   The sequence:$$ 1,2,3,0,0,\ldots $$

   Non-Example:

   The sequence:$$ 1,2,3,4,\ldots $$

4. Evaluation Homomorphism from \(R[x]\) to \(R\)

   Definition:

   An evaluation homomorphism is a function \(\phi_b:R[x]\rightarrow R\), with \(b\in R\). This function is defined by:$$ \phi_b(a_0,a_1,a_2,\ldots) = a_0 + a_1*b + a_2*b^2 + \cdots $$

   Example:

   Non-Example:

5. Zero of a Polynomial

   Definition:

   Let \(p\in R[x]\). \(b\in R\) is a zero of \(p\) if \(\phi_b(p)=0\).

   Example:

   2 is a zero of \((1,-2)\) (i.e. \(x-2\)).

   Non-Example:

   There are no zeroes of the polynomial \((1,0,1) = x^2+1\), in the real numbers.