Math 361, Spring 2018, Assignment 6

Read:

1. Section 27.

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

1. Generator (of the ring $F[x]/\left\langle m\right\rangle$; by default we denote this by $\alpha$).
2. Modulus (of the ring $F[x]/\left\langle m\right\rangle$).
3. Standard representation (of an element of $F[x]/\left\langle m\right\rangle$).
4. Prime ideal.

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

1. Theorem concerning existence and uniqueness of standard representations (of elements of $F[x]/\left\langle m\right\rangle$).
2. Description of the prime ideals of $\mathbb{Z}$.

Solve the following problems:

1. Section 27, problems 1, 3, 5, 7, and 9.
2. Prove that the ring $\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle$ is a field. How many elements does it have?
3. Let $\alpha$ denote the generator of the field $\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle$. Compute the standard representation of the product $(\alpha^2+1)(\alpha^2+\alpha)$.
4. Prove that the ring $\mathbb{Z}_7[x]/\left\langle x^3+2\right\rangle$ is a field. How many elements does it have?
5. (Sieve of Eratosthenes, for integers) Make a list of all integers, in order, from $2$ to $20$. Circle the first number on the list, then cross off all of its multiples. Then circle the next remaining number on the list, and cross off all of its multiples. Continue until all numbers on the list are either circled or crossed off. Which numbers are circled?
6. (Sieve of Eratosthenes, for polynomials) Make a list of all polynomials of degree three or less in $\mathbb{Z}_2[x]$, listing linear polynomials first, then quadratics, then cubics. Circle the first polynomial on the list, then cross off all of its multiples. Then circle the next remaining polynomial on the list, and cross off all of its multiples. Continue until all polynomials on the list are either circled or crossed off. Which polynomials are circled?

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Questions:

Are there any good examples of the fields like R adjoin X or how fields are made up of ideals in cryptography or applied sciences? -Tim

Yes. In elliptic curve cryptography, one needs to do a lot of calculations related to low-dimensional algebraic geometry defined over a finite field. The two most common choices are prime fields (meaning $\mathbb{Z}_p$ for some very large prime $p$) and binary fields (meaning $\mathbb{Z}_2[x]/\left\langle m\right\rangle$ for some irreducible $m\in\mathbb{Z}_2[x]$ of very large degree). Binary fields are of sufficient practical importance that some chip manufacturers have implemented them in hardware.

-Steven.Jackson (talk) 20:20, 8 March 2018 (EST)