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Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite On Mathworld http://mathworld.wolfram.com/PrimeConstellation.html it is first stated that a prime constellation is a sequence of k prime numbers, for which the gap between the last and the first minimizes. But later they show a table with prime constellations and cousin primes (p,p+4) is said to be a prime constellation even though the gap is not minimized because of the 2-tupel (p,p+2). What is the exact definition of a prime constellation and is there some terminology for any sequence of primes whether the gap is minimized or not? I appreciate any clarification because the definition seems to be confusing prime-numbers prime-gaps distribution-of-primes share | cite | improve this question asked Aug 7 at 20:53 Mister Set 494 2 10 I found the OEIS Wiki page on the subject (oeis.org/wiki/Prime_constellations) to have a nice distinction betwen cluster / k-tuple and constellation. ...

Clash Royale CLAN TAG #URR8PPP up vote 15 down vote favorite 7 Suppose $E/F$ is an algebraic extension, where every polynomial over $F$ has a root in $E$. It's not clear to me why $E$ is actually algebraically closed. I attempted the following, but I don't think it's correct: I let $f$ be an irreducible polynomial in $E[X]$. I let $alpha$ be a root in some extension, so $f=m_alpha,E$. Since $alpha$ is algebraic over $E$, it is also algebraic over $F$, let $m_alpha,F$ be it's minimal polynomial. I now let $K$ be a splitting field of $m_alpha,F$, which is a finite extension since each root has finite degree over $F$. If $m_alpha,F$ is separable, then $K/F$ is also separable, so as a finite, separable extension, we can write $K=F(beta)$ for some primitive element $beta$. By assumption, $m_alpha,F$ has a root in $E$, call it $r$. Then we can embed $F(beta)$ into $r$ by mapping $beta$ to $r$. It follows that $m_alpha,F$ splits in $E$. Since $fmid m_alpha,F$, we ...