Expert Answers to Algebra 2 Problems

Showing that $\mathbb{R}[X,Y]/(X^2+1,Y^2+1)\cong <!--\; \cong \; -->\mathbb{C}\times <!--\; \times \; -->\mathbb{C}$
A problem in my textbook asks me to show that $\mathbb{R}[X,Y]/(X^2+1,Y^2+1)\cong <!--\; \cong \; -->\mathbb{C}\times <!--\; \times \; -->\mathbb{C}$. After I've tried solving this for a long time, I looked at the textbook's hint and it said that I should consider the homomorphism $\mathrm{\varphi <!--\; \varphi \; -->}:\mathbb{R}[X,Y]\to <!--\; \to \; -->\mathbb{C}\times <!--\; \times \; -->\mathbb{C}$, $\mathrm{\varphi <!--\; \varphi \; -->}(f)=(f(i,i),f(i,-<!--\; -\; -->i))$ and this would be a surjective ring homomorphism whose kernel is the ideal $(X^2+1,Y^2+1)$, so I would be done.
The fact that this is a ring homomorphism is pretty obvious to me. However, I am really struggling to show that this function is surjective and to find the kernel.
I think that this all stems from the fact that I am not used to the arithmetic properties of polynomial rings in two indeterminates. Anyway, back to the problem. For the surjectivity I tried playing arround with a few polynomials and I hoped to kind of guess the right one, but no succes. Showing that $(X^2+1,Y^2+1)\subset <!--\; \subset \; -->\mathrm{Ker}<!--\; \; -->\mathrm{\varphi <!--\; \varphi \; -->}$ is really straightforward, but the other inclusion is really killing me. So, how should I go about doing this?

If both $a,b>0$, then $a^a{b}^b\ge <!--\; \ge \; -->a^b{b}^a$, if both a and b are positive.

There is a string of red and blue lights strung around a window. The red lights blink every 3 seconds, and the blue lights blink every 5 seconds. How many times win they blink at the same time in one minutes?

Logarithm Question (Find x)
How to solve x for
$x^{2{\log }_{10}<!--\; \; -->x}=\frac{x^5}{100}$?

A corn field with a length of 0.5 mile and a width of 0.25 mile. the farmer harvested 80,000 bushels of corn per square mile and made $19,000 from the harvest. How much did the farmer recieve for each bushel?

How many digits will $a{b}^c$ have?
I know that the digits of $b^c$ is calculated so:
$\lfloor <!--\; \lfloor \; -->c{\log }_{10}<!--\; \; -->b\rfloor <!--\; \rfloor \; -->+1$
but what about $a{b}^c$?

Smallest prime in arithmetic progressions: upper bounds?
Given $N>2$, find a prime p such that
$\begin{array}{}\text{(1)}& p\equiv <!--\; \equiv \; -->1(\mathrm{mod}N).\end{array}$
(Of course, Dirichlet's theorem assures us that infinitely many such p exist.)
Now, I do not know of a clever way to find such a p, so I suggested simply iterating through the arithmetic progression $1+kN$ for $k=0,1,2,\dots <!--\; \dots \; -->$, and terminating when we hit the first prime number. Clearly, this procedure is efficient if and only if the smallest prime p satisfying (1) is at most $N^{O(1)}$. (Edit: This claim is incorrect; see below.) I want to know if such a bound holds or not.
More generally,
Given an integer N, what upper bound do we know on the smallest prime satisfying (1)?
I did a quick check on Wikipedia and Mathworld, but cannot find any relevant results.

Determine whether the equation is exact. If it is, then solve it $(4x^{3}y+6)dx+(x^4-<!--\; -\; -->3)dy=0$

Let $$x=2,\ln <!--\; \; -->y=5,\ln <!--\; \; -->z=16$$
Find the value of $$(\ln <!--\; \; -->(\frac{x}{y})+\ln <!--\; \; -->(\sqrt[4]{xz}))$$

A plane flew 1890 mi in 7 hr with the wind. It would take 9 hr to travel the same distance against the wind. What is the speed of the plane in still air and the speed of the wind?

Modeling Integer LP
In the next semester, the school plans to replace the tutorials by self-organized student groups. Your task is to find the best possible partition of students into groups and to appoint a group leader for each group. To achieve this task, you are given a list of students students. Each student needs to be assigned to exactly one group, and each group needs to have a (student) leader. Every student can be a leader, but we can have only one per group. A leader needs to be assigned to the group he is leading. The number of groups needs to be at least g and at most G. The quality of the groups depends on two criteria. First, the collaboration quality of a student i in a group with leader j is given by collaboration[i,j]. Furthermore, each student i has a certain leadership quality given by leadership[j]. You have to maximize the overall collaboration quality plus the sum of the leadership qualities of the leaders.
I have come up with the following and was wondering if I'm missing something or if someone can show me in the right direction:
I used the variable $x_{i,j}$ if a student i is assigned to the group with leader j ($x_{i,j}=1$) or not ($x_{i,j}=0$). The variable yj shall denote whether student j is a leader ($y_j$) or not ($=0$).
I assumed that there are n students and m leaders. Then I came up with the following LP:
max $\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{x}_{i,j}\cdot <!--\; \cdot \; -->{\text{collaboration}}_{i,j}+\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{y}_j\cdot <!--\; \cdot \; -->{\text{leadership}}_j$s.t.
$\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{x}_{i,j}=1\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,n\}$
$g\le <!--\; \le \; -->\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{y}_j\le <!--\; \le \; -->G$
$x_{i,j}\le <!--\; \le \; -->y_j\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,n\}\mathrm{\forall <!--\; \forall \; -->}j\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,m\}$
$x_{i,j}\in <!--\; \in \; -->\{0,1\}\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,n\}\mathrm{\forall <!--\; \forall \; -->}j\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,m\}$
$y_j\in <!--\; \in \; -->\{0,1\}\mathrm{\forall <!--\; \forall \; -->}j\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,m\}$
This is as far as I have gotten. I know I'm missing the connection between $x_{i,j}$ and $y_j$. Also I'm not sure about my function I want to maximize. Help would be appreciated.

Algorithm to find the irreducible polynomial
What algorithm can be used find an irreducible polynomial of degree n over the field GF(p) for prime p. The reason I ask is I want to make a program for finite field arithmetic, but creating a field $GF(p^k)$ requires finding a irreducible polynomial. (I'm mostly interested in $GF(2^k)$.)

I am reading Fried-Jarden's book Field Arithmetic and I have a question on Pseudo Algebraically closed (PAC) fields.
We say that a field K is PAC if every variety absolutely irreducible over K has a K-rational point. Recall that a variety is absolutely irreducible if it is irreducible over $\stackrel{¯<!--\; ¯\; -->}{K}$, and a K-rational point is an element on the variety with all its coordinates in K.
My question is the following: Is Q a PAC field? And more generally, is any number field PAC?
My attempt: I would like to find a polynomial in two variables with coefficients in Q which is irreducible over C and that does not have any root in ${\mathbb{Q}}^2$. This would show that Q is not PAC, which is my guess, but so far i've been unable to find this example.
I am also guessing that, more generally, every number field is not PAC, although this question is much more difficult...

Confusion with modeling a trigonometric function with phase shift
I am studying trigonometry on Khan Academy and came across this problem:
The moon's distance from Earth varies in a periodic way that can be modeled by a trigonometric function.
When the moon is at its perigee (closest point to Earth), it's about 363,000 km away. When it's at its apogee (farthest point from Earth), it's about 406,000 k away. The moon's apogees occur 27.3 days apart. The moon will reach its apogee on January 22, 2016. Find the formula of the trigonometric function that models the distance D between Earth and the moon t days after January 1, 2016. Define the function using radians.
$D(t)=$
So the steps I took are:
Finding the amplitude: $(406000-<!--\; -\; -->36300)/2=21500$
Finding the midline: $21500+36300=384500$
Figuring out whether to use cosine or sine:
I figured that I can treat January 2nd as the beginning of the year. So I used cosine. Since at 0, a cosine function is at its max value.
4. The period: 1 year is a period so it must be $2\mathrm{\pi <!--\; \pi \; -->}/365$
5. The function without the shift is now: $21500\cos <!--\; \; -->(2\mathrm{\pi <!--\; \pi \; -->}/365t)+384500$
6. Now I must find the value of u in order to properly shift the function. I imagine that this must be $t-<!--\; -\; -->27.3$ since it is 27.3 days after January 1.
I feel like I must be missing something here or got one of the steps wrong. Please guide me in the right direction.

Definition and decidability of bounded quantifiers
Consider quantifier-free formulas $P(x,y)=Q(x,y)$ of Peano arithmetic.
Consider P(x,y),Q(x,y) to be terms composed of variables $x,y,\mathrm{succ},+,\times <!--\; \times \; -->$.
Note that these are equivalent with polynomials $P(x,y),Q(x,y)\in <!--\; \in \; -->{\mathbb{Z}}_{+}[x,y]$.
Let R(x) be correspondingly a term, resp. a polynomial in ${\mathbb{Z}}_{+}[x]$.
The quantifier of a formula $(\mathrm{\exists <!--\; \exists \; -->}y)P(x,y)=Q(x,y)$ is bounded if there is a polynomial R(y) such that
$$(\mathrm{\forall <!--\; \forall \; -->}x)(((\mathrm{\exists <!--\; \exists \; -->}y)P(x,y)=Q(x,y))\to <!--\; \to \; -->(y\le <!--\; \le \; -->R(x)))$$
which is equivalent with
$$(\mathrm{\forall <!--\; \forall \; -->}x)(((\mathrm{\exists <!--\; \exists \; -->}y)P(x,y)=Q(x,y))\to <!--\; \to \; -->((\mathrm{\exists <!--\; \exists \; -->}z)y+z=R(x)))$$
which is a pure PA sentence again.
(Note that the righthand side of $\to <!--\; \to \; -->$ is a only a shorthand!)
I wonder:
Is this the correct definition of a bounded quantifier?
Has the defining sentence to be true or provable?
Is the property of a sentence to have a bounded quantifier decidable?

How do you graph ${\displaystyle f\left(x\right)=\frac{x^2-2x}{x^2-4}}$ using holes, vertical and horizontal asymptotes, x and y intercepts?

Approximating polynomial roots without complex arithmetics
Suppose that we have polynomial of degree greater than two
$P(x)=a_n{x}^n+a_{n-<!--\; -\; -->1}{x}^{n-<!--\; -\; -->1}{+}_{\cdots <!--\; \cdots \; -->}+a_{1}x+a_0$
with real coefficients and we want to approximate all roots of this polynomial without complex arithmetic
It seems to be good idea to get quadratic factors of P(x)
$P(x)=Q(x)(x^2-<!--\; -\; -->px+q)+R(p,q)x+S(p,q)$
Now we want that after each iteration R(p,q) and S(p,q) be closer to zero until given tolerance $\mathrm{\epsilon <!--\; \epsilon \; -->}$
Now we need to solve system of nonliear equations
$$\begin{gathered} R(p,q)=0 \\ S(p,q)=0 \end{gathered}$$
and how to solve this system and present solution in such way it is easy to write code for it

Using laws of logarithms, write the expression below as a single logarithm.
$$5\log <!--\; \; -->(16-<!--\; -\; -->x^2)-<!--\; -\; -->(\log <!--\; \; -->(4+x)+\log <!--\; \; -->(4-<!--\; -\; -->x))$$

Logistic Regression and modelling probabiliy $\mathrm{\pi <!--\; \pi \; -->}$
If the link function is
$$g(\mathrm{\pi <!--\; \pi \; -->})=\log <!--\; \; -->(\frac{\mathrm{\pi <!--\; \pi \; -->}}{1-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}})=x^T\mathrm{\beta <!--\; \beta \; -->}$$
show that this is equivalent to modeling the probability $\mathrm{\pi <!--\; \pi \; -->}$ as,
$$\mathrm{\pi <!--\; \pi \; -->}=\frac{e^{x^T}\mathrm{\beta <!--\; \beta \; -->}}{1+e^{x^T}\mathrm{\beta <!--\; \beta \; -->}}$$
Again I think this is a simply inverse. But I'm not quite sure how to do it.

Natural log problem divide by zero problem for stock/fx contributions
The Stock Price move from 100 ($p_1$) to 150 ($p_2$) and the FX rate moves from 1.2 ($c_1$) to 0.8 ($c_2$). therefore the base currency value stays the same.
I am looking for the fx vs stock contribution over the day using the natural log.
Usually I can use the following formula, but in the above case I get a divide by zero error.
$\frac{p_2{c}_2-<!--\; -\; -->p_1{c}_1}{1+\frac{ln(p_2/p_1)}{ln(c_2/c_1)}}$
fx contribution = $\frac{p_2{c}_2-<!--\; -\; -->p_1{c}_1}{1+\frac{ln(c_2/c_1)}{ln(p_2/p_1)}}$
Any obvious solutions?