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Dec
28
comment How to use a difference table to find a formula for a given sequence?
Both mean multiplication.
Dec
28
comment How to use a difference table to find a formula for a given sequence?
It’s $\verb+\cdot+$.
Dec
28
comment How to use a difference table to find a formula for a given sequence?
Because that’s the mathematicians’ way. It’s just one of the many ways in which the notation of mathematics differs from the notation used by computer folks.
Dec
28
comment An equation over $\Bbb F_{3^k}$
And it’s extremely easy to do computations in the field of nine elements. I recommend to you in the strongest terms the expenditure of a few minutes working things out in this field, using @HenningMakholm’s description. For instance: find an element generating the multiplicative group, and write out its successive powers.
Dec
28
comment How to use a difference table to find a formula for a given sequence?
These are the factorials. So $n!$ is the product of the integers from $1$ through $n$ inclusive: $4!=1\cdot2\cdot3\cdot4=24$, etc.
Dec
28
answered How to use a difference table to find a formula for a given sequence?
Dec
26
comment Question about localization
It’s just saying that the two-step localization, first at $Q$ and then at the smaller ideal $P$, can be done in a single step. I’m feeling too lazy and tired to work it out, but aren’t the fractions the same for the two rings?
Dec
25
comment Subfield of $\mathbb{R}$ such that $\Bbb R/K$ is finite.
@MarcoFlores, no, I think the two facts are independent, even if of a similar flavor. You can prove, by a philosophically similar method, that the $p$-adic fields $\mathbb Q_p$ also have no nontrivial automorphisms.
Dec
25
comment Equivalence of $\pi$ is the first positive zero of the taylor series for $\sin(x)$ and $\pi/4 = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$
@aes, thanks for filling in for my ignorance.
Dec
24
answered How to remember trigonometric ratios for allied angles?
Dec
24
comment Subfield of $\mathbb{R}$ such that $\Bbb R/K$ is finite.
A very natural question, but as @Timbuc’s answer points out, Artin-Schreier forbids such a field. Do you know the proof that $\mathbb R$ has no nontrivial automorphisms? It’s much more basic than A-S, and you can prove it yourself. It demonstrates the weaker proposition that there are no fields $K$ over which $\mathbb R$ is normal.
Dec
24
comment Equivalence of $\pi$ is the first positive zero of the taylor series for $\sin(x)$ and $\pi/4 = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$
I’m not an analyst, but my understanding is that it’s a moderately deep result that the endpoint-evaluation of the series for $\arctan(x)$ gives $\arctan(1)$.
Dec
24
answered $A/B \cong A \iff B = \{e\}$ for groups $A,B$?
Dec
23
comment The finite field extension
This is really the best way of proving it.
Dec
23
comment What's the relation between factors of a number and its square root?
Have you done the corresponding computations for all integers, say, between $10$ and $200$? Insight comes from experience; maybe you’ll find something wonderful, and maybe you’ll find that your idea doesn’t amount to anything. This is how mathematics goes.
Dec
23
comment If $G$ is a finite group and $H$ is a subgroup of $G$, then $|H| = |gH|$ for every $g \in G$.
This shows clearly why a conceptual approach is so much better than one that thinks of the sets in question as lists of elements. It’s quicker and cleaner, and it explains better.
Dec
23
answered Notations: use of parentheses with “mod” and the “|” symbol
Dec
23
comment Showing $\mathbb Z[\alpha]/\alpha \mathbb Z[\alpha] \cong O_K/\alpha O_K$
Maybe so, but still more efficiently than I would have done it.
Dec
22
comment What is the name of the group that contains both lines and shapes?
Not everything in mathematics has its own name. Why don’t you just call them objects?
Dec
22
answered Superfunctions with complex iteration indices: Interpretation