Math 361, Spring 2017, Assignment 3

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

1. Polynomial function defined by a polynomial expression.
2. Zero (a.k.a. root) of a polynomial.
3. Degree (of a polynomial).
4. $R[x, y]$.
5. $R[x_1,\dots,x_n]$.

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

1. Bound on the degree of the sum of two polynomials.
2. Bound on the degree of the product of two polynomials.
3. Exact formula for the degree of the product of two polynomials, in case the base ring is an integral domain.

Solve the following problems:

1. Section 22, problems 5, 7, 9, 11, 13, and 15.
2. Prove that if $D$ is an integral domain, then so is D[x]. (Hint: the only hard part is showing that the product of two non-zero polynomials is non-zero. So suppose $f,g\neq0$. What can you say about $\mathrm{deg}(f)$, $\mathrm{deg}(g)$, and $\mathrm{deg}(fg)$? Alternatively, what can you say about the highest-degree term of $fg$?)
3. Let $D$ be any integral domain. The field of rational expressions in one indeterminate, with coefficients in $D$ is the field $D(x) = \mathrm{Frac}(D[x])$. Write down two or three sample elements of $\mathbb{Q}(x)$, then show how to add, multiply, and invert them.
4. More generally, the field of rational expressions in $n$ indeterminates is the field $D(x_1,\dots,x_n) = \mathrm{Frac(D[x_1,\dots,x_n])}$. Give sample elements and sample calculations, as above, for the field $\mathbb{Q}(x,y)$.
5. Let $p$ be any prime number. Show that $\mathbb{Z}_p(x)$ is always an infinite field (even though $\mathbb{Z}_p$ itself is a finite field).

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==Questions:== I have no idea where to start with Question 5 in section 22. How many polynomials are there of degree $\leq 3 \in \mathbb{Z}_2[x]$? include (0) Any suggestions I have been looking at this problem for a few hours now still have no clue.

A polynomial of degree three or less has the form $a_0 + a_1x + a_2x^2 + a_3x^3$ with the coefficients coming from $\mathbb{Z}_2$. How many choices are there for each coefficient? -Steven.Jackson (talk) 18:00, 13 February 2017 (EST)

Solutions: Question 11) $\phi_4(3x^{106}+5x^{99}+2x^{53})$ in $\mathbb{Z}_7$

1) First use Fermat's theorem. $a^{p-1}= 1(mod$ $p$) in our case $a^{7-1}=1 (mod$ $7$) So $a^6 = 1 (mod$ $7$)

2)Now take each power in the polynomial function $(3x^{106}+5x^{99}+2x^{53})$ So you get $106=6 \cdot 17+4$, $99=6 \cdot 16 + 3$, $53=6 \cdot 8 +5$

3) Now plug in 4 into the new x and the new exponent powers. so that you get $3 \cdot 4^4(4^6)^{17} + 5 \cdot 4^3(4^6)^{16} + 2 \cdot 4 ^5 \cdot (4^6)^8$

$3 \cdot 4^4(1)^{17} + 5 \cdot 4^3(1)^{16} + 2 \cdot 4 ^5 \cdot (1)^8$

$3 \cdot 4^4 + 5 \cdot 4^3 + 2 \cdot 4 ^5$

$3 \cdot 256 + 5 \cdot 64 + 2 \cdot 1024$

$ 768+320+2048$

$3136$

$\frac{3136}{7}$=448 Remainder of 0

thus the answer is zero

My question for this problem is there something I am missing that would make less step. Specify is their a way to make this line $3 \cdot 4^4 + 5 \cdot 4^3 + 2 \cdot 4 ^5$ be simple without doing out by hand or without a calculator?

The powers of 4 aren't too hard to compute by hand (in $\mathbb{Z}_7$) if you reduce after each multiplication to keep the numbers small. So: $4^1 = 4$, then $4^2 = 4\times 4 = 16 = 2$, then $4^3 = 4 \times 4^2 = 4 \times 2 = 8 = 1$. Then the powers repeat, so $4^4 = 4$, and $4^5 = 2.$ -Steven.Jackson (talk) 18:00, 13 February 2017 (EST)