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 Carroll, 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 \(\phi_b(p)=0\). This is really just a way to formalize the normal idea of a root.

   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.

Theorems:

1. Universal Mapping Property of the Field of Fractions

   Let \(D\) be an integral domain with field of fractions \(F\). If there is another field \(E\) such that \(D\) can be mapped into \(E\), meaning there is a monomorphism \(\phi:D\rightarrow E\), then there is another monomorphism \(\psi:F\rightarrow E\) defined by \(\psi((a,b)) = \phi(a)\phi(b)^{-1}\).

2. Universal Mapping Property of \(R[x]\)

   Let \(R\) and \(S\) be rings, with a ring homomorphism \(\phi:R\rightarrow S\). Let \(a\in S\). Then there is a unique homomorphism \(\psi_a:R[x] \rightarrow S\) such that:$$ \psi_a(x) = a\\ \psi_a(c) = \phi(c) $$

   Specifically, if we let \(p=(p_0,p_1,\ldots)\in R[x]\), then \(\psi_a(p)\) must be:$$ \psi_a(p) = \phi(p_0) + \phi(p_1)a + \phi(p_2)a^2+\cdots $$

   So \(\psi_a\) basically translates a polynomial over \(R[x]\) to a corresponding polynomial over \(S[x]\) and then evaluates at \(a\).

3. Relationship Between \(R[x]\) and the Ring of Polynomial Functions on \(R\)

   There is an epimorphism from \(R[x]\) to the ring of polynomial functions. If we define \(\phi:R[x]\rightarrow Fun(R,R)\) by:$$ \phi(p) = p(x) = p_0 + p_1x + p_2x^2 + \cdots $$

   Then \(\phi\) is an epimorphism but not a monomorphism. Two different formal polynomials can result in the same polynomial function, but every polynomial function can be formed from a formal polynomial.