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1500 questions
72
votes
2 answers
Number of monic irreducible polynomials of prime degree $p$ over finite fields
Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$? Thanks!
IBS
- 4,375
72
votes
7 answers
How to find the factorial of a fraction?
From what I know, the factorial function is defined as follows: $$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$ And $0! = 1$. However, this page seems to be saying that you can take the factorial of a fraction, like, for instance, $\frac{1}{2}!$, which they…
Cisplatin
- 4,855
72
votes
6 answers
"Gaps" or "holes" in rational number system
In Rudin's Principles of Mathematical Analysis 1.1, he first shows that there is no rational number $p$ with $p^2=2$ . Then he creates two sets: $A$ is the set of all positive rationals $p$ such that $p^2<2$ , and $B$ consists of all positive…
Larry
- 761
72
votes
7 answers
Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?
I self-realized an interesting property today that all numbers $(a,b)$ belonging to the infinite set $$\{(a,b): a=(2l+1)^2, b=(2k+1)^2;\ l,k \in N;\ l,k\geq1\}$$ have their AM and GM both integers. Now I wonder if there exist distinct real numbers…
Gaurang Tandon
- 6,675
72
votes
4 answers
A nasty integral of a rational function
I'm having a hard time proving the following $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1} \, dx = \frac{\pi}{2}.$$ Mathematica has no problem evaluating it while I haven't the slightest…
user54031
72
votes
2 answers
Is there an intuitive reason for a certain operation to be associative?
When I read Pinter's A Book of Abstract Algebra , Exercise 7 on page 25 asks whether the operation $$x*y=\frac{xy}{x+y+1}$$ (defined on the positive real numbers) is associative. At first I considered this to be false, because the expression is so…
Zirui Wang
- 1,567
72
votes
12 answers
What is the largest eigenvalue of the following matrix?
Find the largest eigenvalue of the following matrix $$\begin{bmatrix} 1 & 4 & 16\\ 4 & 16 & 1\\ 16 & 1 & 4 \end{bmatrix}$$ This matrix is symmetric and, thus, the eigenvalues are real. I solved for the possible eigenvalues and, fortunately, I found…
Cloud JR K
- 2,586
72
votes
4 answers
Center-commutator duality
I'm reading this article by Keith Conrad, on subgroup series. I'm having trouble with a statement he does at page 6: Any subgroup of $G$ which contains $[G,G]$ is normal in $G$ . He says this as evidence that commutator and center play dual roles,…
Bruno Stonek
- 13,319
72
votes
11 answers
How can I find the points at which two circles intersect?
Given the radius and $x,y$ coordinates of the center point of two circles how can I calculate their points of intersection if they have any?
Joe Elder
- 729
72
votes
2 answers
Let, $A\subset\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$.
A challenge problem from Sally's Fundamentals of Mathematical Analysis. Problem reads: Suppose $A$ is a subset of $\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$ with the usual…
David Bowman
- 5,838
- 1
- 18
- 30
72
votes
11 answers
Why is the complex plane shaped like it is?
It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someone please explain the logic or rationale behind this?…
user64742
- 2,226
72
votes
18 answers
Why is radian so common in maths?
I have learned about the correspondence of radians and degrees so 360° degrees equals $2\pi$ radians. Now we mostly use radians (integrals and so on) My question: Is it just mathematical convention that radians are much more used in higher maths…
Slater
- 795
72
votes
3 answers
Why is the Daniell integral not so popular?
The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some serious flaws. The most natural way to fix all the…
gifty
- 2,321
72
votes
11 answers
Have there been efforts to introduce non Greek or Latin alphabets into mathematics?
As a physics student, often I find when doing blackboard problems, the lecturer will struggle to find a good variable name for a variable e.g. "Oh, I cannot use B for this matrix, that's the magnetic field". Even ignoring the many letters used for…
Mark Allen
- 833
72
votes
8 answers
Why casting out $9$'s and $11$'s works in divisibility tests (similarly for other bases)
I don't know if this is a well-known fact, but I have observed that every number, no matter how large, that is equally divided by $9$ , will equal $9$ if you add all the numbers it is made from until there is $1$ digit. A quick example of what I…
JD Isaacks
- 883