Is there an analogue of finite fields for products of two prime powers? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:00:31Z http://mathoverflow.net/feeds/question/85480 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85480/is-there-an-analogue-of-finite-fields-for-products-of-two-prime-powers Is there an analogue of finite fields for products of two prime powers? Alan Haynes 2012-01-12T10:53:36Z 2012-01-14T15:50:06Z <p>The collection of prime powers can be characterized in the following way:</p> <blockquote> <p>There is a field with $q$ elements if and only if $q$ is a prime power. Furthermore if it exists then this field is unique up to isomorphism.</p> </blockquote> <p>My question is,</p> <blockquote> <p>Is there an analogous characterization for positive integers which are products of two distinct prime powers?</p> </blockquote> <p>To avoid triviality let us say that the characterization should be presented independently of the prime factorization of the number involved. So we are looking for a statement like,</p> <blockquote> <p>"There exists a set from a certain class which has property $P(n)$ if and only if $n$ is a product of two distinct prime powers,"</p> </blockquote> <p>and <em>both the set from the relevant class and the property $P(n)$ should be defined in a way that does not inherently depend on the prime factorization of $n$.</em></p> http://mathoverflow.net/questions/85480/is-there-an-analogue-of-finite-fields-for-products-of-two-prime-powers/85494#85494 Answer by Goldstern for Is there an analogue of finite fields for products of two prime powers? Goldstern 2012-01-12T14:53:06Z 2012-01-12T14:53:06Z <p>For any (first order, but other variants are also reasonable) formula $\varphi$ without free variables, finite model theory defines $S(\varphi)$, the "spectrum" of $\varphi$, as the set of all positive natural numbers $n$ such that there exists a structure of size $n$ satisfying $\varphi$. </p> <p>(As far as I remember, there is no nice characterization of those subsets of $\mathbb N$ which are spectra. In particular: it is still open whether the set of all spectra of first order formulas is closed under complements.)</p> <p>It would be reasonable to define $S^*(\varphi)$ as the set of all $n$ such that there is a unique (up to isomorphism) structure of size $n$ satisfying $\varphi$. </p> <p>As Emil JeÅ™ábek has implicitly pointed out in his first comment, there is a first order formula $\varphi_{\text{product of fields}}$ such that $S^*(\varphi_{\text{product of fields}} )=S(\varphi_{\text{product of fields}} ) =$ the set of all products of two prime powers. (The formula is really quite explicit; I do not give it here as it would not add any relevant information to my answer, I think.) </p> <p>I know that this answer is in a sense trivial. But I don't see a formal criterion that will distinguish the trivial from the nontrivial answers. </p> http://mathoverflow.net/questions/85480/is-there-an-analogue-of-finite-fields-for-products-of-two-prime-powers/85531#85531 Answer by Kevin O'Bryant for Is there an analogue of finite fields for products of two prime powers? Kevin O'Bryant 2012-01-12T22:01:35Z 2012-01-13T07:30:58Z <p>There are at most 2 groups (up to isomorphism) of order $n$, and there is not a field of order $n$, if and only if $n$ is the product of two distinct primes.</p> <p>This doesn't cover "prime powers", but at least it's nontrivial.</p> http://mathoverflow.net/questions/85480/is-there-an-analogue-of-finite-fields-for-products-of-two-prime-powers/85546#85546 Answer by Noam D. Elkies for Is there an analogue of finite fields for products of two prime powers? Noam D. Elkies 2012-01-13T02:07:46Z 2012-01-13T02:07:46Z <p>If nothing else then $n$ is a product of powers of two distinct primes <strong>iff</strong> there is a unique pair of relatively prime natural numbers, other than the trivial $\lbrace 1,n \rbrace$, whose product is $n$.</p> http://mathoverflow.net/questions/85480/is-there-an-analogue-of-finite-fields-for-products-of-two-prime-powers/85671#85671 Answer by felix for Is there an analogue of finite fields for products of two prime powers? felix 2012-01-14T15:50:06Z 2012-01-14T15:50:06Z <p>A natural number $n$ is the product of precisely two prime powers if and only if there exists an abelian group of order $n$ having precisely two maximal subgroups. (And that group is unique up to isomorphism.)</p>