Math 480, Spring 2013, Assignment 1
From cartan.math.umb.edu
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:[edit]
- 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):[edit]
- Theorem on vanishing polynomial functions over an infinite field (Proposition 5 in the text).
Questions:[edit]
- 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
So this was a trick question. Eventually, you would come across the term \((x-x)\), which of course is zero, making \(P(x) = 0 + 1 = 1\)
- 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)
- Thank you for your quick response! My second question is in regard to question 3. In the example from class, we used Z2, and showed how a polynomial with coefficients in Z2 could evaluate to zero for any given possible combination of inputs. Is this the same concept that we would apply to solve problem number three? Thank you - --Rob Moray
- Yes, it's possible to find a polynomial that vanishes for any possible input. But it won't be the same polynomial as in class. (Actually my example was unnecessarily complicated; one can find low-degree examples in a single variable.) Steven.Jackson (talk) 07:45, 2 February 2013 (EST)
- Ahh, so with a single variable, we would just assume the other inputs would not be evaluated, as there would be no variable to associate with them. For instance, if we had, for some polynomial \(p\), \(p=x_{1}^2\), then even if I evaluated \(\hat{p}(0,1,2)\), I would still get \(\hat{p}(0,1,2)=0^2=0\) because there would be no \(x_2\) or \(x_3\) to plug the 1 and 2 into. I hope I am going in the right direction here. - --Rob Moray
- Yes, that's right. If \(n=3\) and \(p=x_1^2\) then \(\hat{p}(0,1,2)=0\). But since I didn't specify a value for \(n\) you are free to set it as low or as high as you like. I happen to know that there are low-degree examples of this phenomenon with \(n=1\), so in order to keep things simple I was steering you in that direction. But you may use whatever number of variables you wish. --Steven.Jackson (talk) 11:36, 2 February 2013 (EST)
- Ahh, so with a single variable, we would just assume the other inputs would not be evaluated, as there would be no variable to associate with them. For instance, if we had, for some polynomial \(p\), \(p=x_{1}^2\), then even if I evaluated \(\hat{p}(0,1,2)\), I would still get \(\hat{p}(0,1,2)=0^2=0\) because there would be no \(x_2\) or \(x_3\) to plug the 1 and 2 into. I hope I am going in the right direction here. - --Rob Moray
- Yes, it's possible to find a polynomial that vanishes for any possible input. But it won't be the same polynomial as in class. (Actually my example was unnecessarily complicated; one can find low-degree examples in a single variable.) Steven.Jackson (talk) 07:45, 2 February 2013 (EST)
