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 Apr 30 answered Does the PNT establish a connection between primes and the logarithm? Apr 17 comment $f(x) = x^2 + bx + a$ irreducible over $\Bbb F_p$ (finite field of $p$ prime elements) iff $(b^2 - 4a)^{\frac{p-1}{2}} = -1$ in $\Bbb F_p$ First, you need $p \not= 2$. Granting that, show for nonzero $d$ in $\mathbf F_p$ that if $d$ is a square then $d^{(p-1)/2} = 1$ and if $d$ is not a square then $d^{(p-1)/2} = -1$. This is a standard theorem about squares and nonsquares mod $p$ and can be found in almost every number theory book. Look up Euler's congruence. There is no need to use $\mathbf F_{p^2}$. Apr 9 comment If a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a − b, i.e. m | a − b. What is your definition of $a \equiv b \bmod m$? Some would say you're being asked to prove a definition, which doesn't make sense. Apr 4 awarded Good Question Apr 3 comment “Logarithmic derivative of a p-adic number” See fen.bilkent.edu.tr/~franz/publ/hil.pdf, which is the introduction to the English translation of Hilbert's Zahlbericht. Search in this document for the term "logarithmic." You'll find a discussion of Kummer's work on higher reciprocity laws and a reference to Section 131 of the Zahlbericht. Apr 3 comment “Logarithmic derivative of a p-adic number” See Section 13.7 of Washington's Introduction to Cyclotomic Fields (2nd edition). The idea is that certain $p$-adic units $u$ may be special values of a $p$-adic power series, so the logarithmic derivative of that series at the same point is the "logarithmic derivative" of $u$. Of course there's a certain ambiguity here (what if $u$ is a special value of another $p$-adic power series at similar points?), but to see how this is handled just look in Washington's book. Apr 3 awarded Nice Answer Apr 2 comment What does the symbol |_ mean? Какую книгу Вы читаете? Apr 2 answered What does the symbol |_ mean? Mar 26 comment Fining the radius of convergence of $\sum_{k=1} ^{\infty} \frac{2^k*z^{2k}}{k^2+k}$ The series has $z^2$ everywhere, so it converges for $|z^2| < 1/2$. Mar 20 comment Proportion of elements of prime order $p$ in $S_n$ Hey, you deleted one of your comments. You shouldn't do that when other comments are made in response to them, since it causes the discussion to read awkwardly. I wrote that you checked $p=3$ only up to $n=6$, but the earlier comment where you said you did that is now missing, so it appears now that I'm omniscient or something. Mar 20 comment Proportion of elements of prime order $p$ in $S_n$ I advise you to first try to determine the error yourself before asking others to do that, e.g., take $p = 2$ and figure out why your formula is wrong when $n = 6$. Mar 20 revised Proportion of elements of prime order $p$ in $S_n$ added 117 characters in body Mar 20 comment Proportion of elements of prime order $p$ in $S_n$ By the way, in your first comment, using $S_n$ for a number when you're talking about something involving the symmetric group is pretty bad notation. Mar 20 comment Proportion of elements of prime order $p$ in $S_n$ You are not checking far enough. For $p=2$ your formula breaks down starting at $n=6$, for $p=3$ it breaks down starting at $n=8$ (you only say you checked up to $n=6$), and for $p=5$ it breaks down starting at $n=12$. You'll have to figure out yourself why your formula is wrong. Presenting a formula that doesn't work and then asking others why it doesn't work when you never explained where your formula came from is not reasonable. Mar 20 revised Proportion of elements of prime order $p$ in $S_n$ added 140 characters in body Mar 20 answered Proportion of elements of prime order $p$ in $S_n$ Mar 15 comment Polynomial ring is not PID Mar 13 comment Why is Lebesgue so often spelled “Lebesque”? This reminds me of hearing that the name Artin was originally from Armenia, where it was Artinian. So if the common name ending -ian had not been removed when the ancestors of Emil Artin moved to Germany, we might today speak about artinianian modules. Mar 11 comment What is the importance of knowing if a series converges or diverges? Why does this explain the importance of knowing whether a series converges? It doesn't describe any application to a problem not already about infinite series themselves. It's like saying convergence of a sequence can be used to estimate the limit. That doesn't really address why you'd want to do it in the first place.