Math 480, Spring 2013, Assignment 1
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Revision as of 16:31, 1 February 2013 by Steven.Jackson (talk | contribs)
The beginner ... should not be discouraged if ... he finds that he does not have the prerequisites for reading the prerequisites.
- - P. Halmos
Solve the following problems:
- Name two familiar arithmetic structures that are fields, and two familiar arithmetic structures that are not.
- Find an integer \(n\) for which the structure \(\mathbb{Z}_n\) is not a field. Write the multiplication table for this structure, and use the table to explain why, for this choice of \(n\), \(\mathbb{Z}_n\) is not a field. Then choose another \(n\) for which \(\mathbb{Z}_n\) is a field, and use the resulting multiplication table to verify this.
- Find a non-zero polynomial with coefficients in \(\mathbb{Z}_3\) whose associated function vanishes identically.
- What is the degree of the polynomial \(x^3 + x^2yz\)?
- Give an example of a field which is not algebraically closed, and prove that it is not.
Carefully state the following theorems (you need not prove them):
- Theorem on vanishing polynomial functions over an infinite field (Proposition 5 in the text).
Questions:
- What's the degree of this polynomial and how many terms does it have?
- \[P(x) = (x-a)(x-b)...(x-z) + 1\]
- - --Matthew Lehman
- Are we assuming there is an \(x\) constant in there? - --Frank C (talk) 06:08, 1 February 2013 (EST) "06:08, 1 February 2013 (EST)"
- I feel like this is a stupid question, but can we think of a set, like the set of real numbers for instance, as an arithmetic structure? - --Rob Moray
- Not stupid at all---I never defined this term. The short answer is yes. The long answer is that when I say "arithmetic structure," what I really mean is ring: a set of objects (for example, the set of real numbers) together with two binary operations (for example, ordinary addition and multiplication) which satisfy a rather long list of axioms. See Wikipedia:Ring (mathematics) for the full definition and many examples. Steven.Jackson (talk) 11:31, 1 February 2013 (EST)
