{"id":580,"date":"2008-01-16T15:53:49","date_gmt":"2008-01-16T15:53:49","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2008\/01\/16\/fields-characteristics-and-prime-numbers\/"},"modified":"2008-01-16T15:53:49","modified_gmt":"2008-01-16T15:53:49","slug":"fields-characteristics-and-prime-numbers","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2008\/01\/16\/fields-characteristics-and-prime-numbers\/","title":{"rendered":"Fields, Characteristics, and Prime Numbers"},"content":{"rendered":"

When we start looking at fields, there are a collection
\nof properties that are interesting. The simplest one – and
\nthe one which explains the property of the nimbers that
\nmakes them so strange – is called the
\ncharacteristic<\/em> of the field. (In fact, the
\ncharacteristic isn’t just defined for fields – it’s defined
\nfor rings as well.)<\/p>\n

Given a field F, where 0F<\/sub> is the additive
\nidentity, and 1F<\/sub> is the multiplicative identity,
\nthe characteristic of the field is 0 if and only if no
\nsequence of adding 1F<\/sub> to itself will ever result
\nin 0F<\/sub>; otherwise, the characteristic is the
\nnumber of 1F<\/sub>s you need to add together to get
\n0F<\/sub>.<\/p>\n

That sounds confusing – but it really isn’t. It’s just
\nhard to write in natural language. A couple of examples will
\nmake it clear.<\/p>\n

<\/p>\n

The field of nimbers has characteristic two – because
\n1+1=0 in the nimbers – so adding to 1s together gives you
\n0.<\/p>\n

The field of numbers modulo-7 is a nice simple field,
\nwith characteristic 7: in mod-7, you need to add 1+1=2;
\n2+1=3; 3+1=4; 4+1=5; 5+1=6; 6+1=0. So 1+1+1+1+1+1+1=0.<\/p>\n

The real numbers have characteristic 0, because adding
\none repeatedly will never give you 0. In fact, the
\ncharacteristic is 0 for any ordered<\/em> field O – that
\nis, a field with a properly defined “&lt” operator (one
\nwhere ∀x∈O, ∃y∈O : x<y ∧
\n∃z∈0 : z<y, and < is total, transitive,
\nantisymmetric, and antireflexive).<\/p>\n

See? Not hard, right?<\/p>\n

What makes nimbers so strange is that they’re an
\ninfinite set with characteristic two. Our intuitions about
\nnumbers all rely on the structure of an infinite set with
\ncharacteristic 0. All of the strange properties of nimbers
\ncan be traced back to the fact that their basic structure is
\nstrange; and the root of that strangeness always traces back
\nto the field characteristic.<\/p>\n

There’s an interesting notion also associated with the
\ncharacteristic of a field. Like groups could have
\nsub-groups, fields can have sub-fields: a sub-field of a
\nfield (F,+,×) is a field (G,+,×) where G⊆F.
\nFor every field, there is a minimal subfield – a field where
\nremoving any elements would require removing 1 to maintain
\nclosure. The minimal subfield of a field, F, is called the
\nprime subfield<\/em> of F.<\/p>\n

Here’s where we get something fascinating. There are
\nnumerous limits to the kinds of structures that can fulfill
\nthe field axioms to become proper fields. One of the amazing
\nones (at least to me) is that every field has either
\ncharacteristic 0 (in which case it’s isomorphic to a
\nsub-field of the complex numbers), or it has a
\nprime<\/em> characteristic, and it’s prime subfield is
\nisomorphic to a finite field whose size equals it’s
\ncharacteristic.<\/p>\n

What does this tell us? Well, looking at it
\nphilosophically, it means that the prime numbers are very
\ndeeply embedded in the structure of algebra. Even if we
\nthrow away our standard number system and go into the realm
\nof the abstract, the numbers that are prime in the basic set
\nof natural numbers remain fundamental. Both the concept and
\nthe specific values of the prime numbers are deeply embedded
\nall the way down in the foundations of what we know as
\nmathematics. Even when we start from scratch, with abstract
\nsets, and build upwards into mathematical structures, when
\nwe get to the point where we can build things that behave
\neven just a little bit like numbers, the prime numbers are
\ninevitable – even though they aren’t necessarily prime in
\nevery field!<\/p>\n

After all, think about the nimbers! The number 11 for example, is clearly prime in the natural numbers. But in nimbers, it’s 8×4: and yet, even if we play with the nimbers, we’ll wind up finding out that 11 is prime.<\/p>\n

Incidentally, I’m still trying to hack together an implementation of the nimbers. As one of the commenters (Xanthir?) mentioned, it’s hard to get them right, and I haven’t had much free time to try it. I’ve got a bunch of Scheme code that’s close, but it’s still got one recursive case that never terminates. Nimbers are truly annoying.<\/p>\n","protected":false},"excerpt":{"rendered":"

When we start looking at fields, there are a collection of properties that are interesting. The simplest one – and the one which explains the property of the nimbers that makes them so strange – is called the characteristic of the field. (In fact, the characteristic isn’t just defined for fields – it’s defined for […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[26],"tags":[],"class_list":["post-580","post","type-post","status-publish","format-standard","hentry","category-group-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-9m","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/580","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=580"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/580\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=580"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=580"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=580"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}