Class Note for MATH 112 at UA 2
Class Note for MATH 112 at UA 2
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Date Created: 02/06/15
KRONECKER WEBER FOR KIDS 1 GOD CREATED THE INTEGERS AND ALL ELSE WAS THE WORK OF MAN I remember that this famous quote by Kronecker made no sense to me when I saw it on the pages of my Math 112 notes It seemed out of place in a world of limits7 complex numbers and power series7 not to mention how atheistic it sounded For me it took several years to get a coherent picture of what Kronecker probably had in mind In this handout I attempt to explain a tiny bit of 77the work of man77 by exhibiting a special case of the famous Kronecker Weber Theorem Of course7 we have to start with what God has given us7 namely the integers That seems fairly easy7 but soon one realizes that7 as Jerry Shurman once put it7 77The integers are simply too small77 Even worse7 the eld of fractions of the integers7 the rationals7 is essentially as small as the integers Even simple algebraic relations like x2 7 2 0 are not satis ed by any rational number But man7 being the inquisitive creature that its nature entails7 found a way around this by imagining that there were rational number with unusual properties Moreover these new 77rationals77 had 77integers77 of their own Not surprisingly man7s imagination was outrunning his means and he soon realized that there was no use of all that He had created something too big7 in fact immeasurably bigger than the numbers that God had given him7 so he had to rethink his approach and focus on what was manageable and nite So he created Algebraic Number Theory But7 as it often happens7 the new numbers that man created had questions of their own 2 TWO PAGES OF DEFINITIONS AND THEOREMS Since the goal of this handout is to prove in a reasonable amount of detail a special case of the Kronecker Weber Theorem we have to run through a lot of de nitions and theorems most of which for us will be just facts because we wont prove them before we get to the interesting part Algebraic Number Theory is the study of nite extensions of Q All of the de nitions and theorems that follow can be found in Mar or Rib 1 2 KRONECKER WEBER FOR KIDS DEFINITION A complex number will be called an algebraic num ber if it is algebraic over Q and the set of all such numbers will be denoted by Q DEFINITION A complex number is an algebraic integer iff it is a root of some monic polynomial with coef cients in Z The set of such number will denoted by Z Q THEOREM Z C Q C C Moreover Z is a ring and Q is a eld D It is not too dif cult to see that Q cannot be nite and so we wouldn7t have powerful tools under our belt such as induction or linear algebra when we need them So we have to contend ourselves with what are called number elds DEFINITION A number eld is a sub eld of C having nite degree over Q Q Since this implies that all elements of a number eld are algebraic over Q we have that any such eld is a sub eld of The ring that is the intersection of a number eld with Z is the analogue of Z in Q in the sense that Q Z Z DEFINITION For any number eld L the ring LZ Z L is called the ring of integers in L and such rings will be called number rings Q However there is one way in which Zdoes not appear to be analogous to Z This is unique factorization of elements into irreducibles For example in we have 6 2 3 4 7 This in itself does not show that factorization is not unique but using an argument involving some basic Algebraic Number Theory it can be shown that in fact everything in sight except 6 is irreducible It turns out that in the case of number rings we have something which is almost as good Namely unique factorization into prime ideals THEOREM Let LZ be a number ring Every non zero ideal of LZ is uniquely expressible as a product of prime ideals ie for any ideal I of LZ we have n 111282 i1 for some prime ideals Ps and positive integers e D Returning to our original example one can compute lt6gt lt2gtlt3gt lt2 mm 4 mm 2 mu KRONECKER WEBER FOR KIDS 3 and all the ideals on the right are prime in xEM7 which incidentally is Mm but this may not be the case in general We will leave things like the additive structure and the multiplicative structure of LZ and L aside and focus on the following question If we have two number elds F Q L and p E FZ what can we say about the factorization of the ideal p in LZ We have used the concepts of an ideal and element as if they are the same7 and we will continue to do so with the hope that this won7t cause any conclusion Note that when we say the ideal p of LZ7 we mean LZ To sum up7 until further notice 77a prime77 will mean a prime ideal THEOREM Let L and F be number elds Let p be a prime of FZ Z O F7 and P a prime of LZ Z O L Then the following are equivalent 1 LZpQP7thatisPlLZp 21QP 3 P Fzp 4 P Fp B When any of the above conditions holds7 we will say that P lies over p and p lies under P THEOREM Every prime P of LZ lies over a unique prime p of FZ every prime p of FZ lies under some prime P of LZ D The following de nitions and theorem are crucial note that they make sense because of the previous theorem DEFINITION Suppose that p Hf Pfi Then 1 g is called the decomposition number of p in LF 2 for every i I7 9 el is called the rami cation indecc of Pi 3 the dimension fl of LZPZ over FZp is called the inertial degree of P1 in LF lt1 THE DECOMPOSITION EQUATION THEOREM Let n L F7 then with the above notation 9 i1 DEFINITION Suppose that p 191 Pfi7 n L Then if g n we say that p splits completely in LF7 on the other hand if el gt 1 for some i E 1 79 then we say that p rami es or equivalently is rami ed in LZ lt1 4 KRONECKER WEBER FOR KIDS In the case of LF being normal and therefore Galois since we are working over Q and consequently separability is not an issue we can say even more TRANSITIVITY OF THE GALOIS ACTION ON IDEALS If P and P are any prime ideals of LZ such that P FZ P FZ 31 Q then there exist 039 E 9 such that 0P P Proof Let g 01Un GalLF Assume that P 31 TlP for every 0 E 9 Then since TlP is a prime ideal we can use the Chinese Remainder Theorem to nd an x E LZ such that 20 mod P x21 modalP 2391n Let a Halx then from Galois Theory we know that a E FZ and a E P Hence a E FZ P p for some prime ideal p of FZ But then a E FZ P and therefore a E P However 1 P since 0z P for 239 1 n otherwise z flaw E U 1P for some index 239 We have obtained a contradiction and hence there exists 0 E 9 such that TlP P D And we have THE GALOIS DECOMPOSITION EQUATION If LF is normal ie Galois of degree n ifp Hf P in LZ and if KZPl FZp fi then 61 62 69 f1 f9 Moreover if FZp is a nite eld then the LZPZ are isomorphic Proof Let p Hf Pfi For any 039 E 9 we have 11 But for any P 1 gj S 9 there exists 739 E 9 so that 7P1 Pj Now applying unique factorization we get 61 6739 for all j 1 9 Similarly we have LzPj LzTP1 g LzPl andthus f1 fj for allj 1g The last statement in the theorem follows from the uniqueness of nite elds D KRONECKER WEBER FOR KIDS 5 After seeing Galois Theory do all the work for us we need to de ne an important invariant of a number eld called the discriminant Let 01 0 be the automorphisms of L For any n tuple of ele ments 11 an E L de ne the discriminant 6 of a1 an to be the square of the determinant of the matrix with ijth entry Tlaj i e 5117 7an llamaNZ THEOREM For any number eld L there exist element 11 an E LZ that form both a basis for L over Q and a basis for LZ over Z D THEOREM For any number eld L the discriminant of an integral basis is invariant under a change of integral basis and is called the discriminant of L and denoted by 6L D It is not too dif cult to see that 6L 6 Z Finally7 we come to an important theorem7 which will be used repeatedly THEOREM Let p be a prime in Z then p rami es in a number ring LZ ltgt p l 6L D With this our summary of facts will end7 but the reader who wants to learn more about other questions and developments in Algebraic Number Theory can read about them in Mar and Rib 3 THE EASY PART OF KRONECKER WEBER We have already proved the Fundamental Theorem of Cyclic Exten sions in class7 in this section we will prove a special case of what can be called the Fundamental Theorem of Abelian Extensions7 namely THE KRONECKER WEBER THEOREM lf L is a number eld7 which is an Abelian extension of Q7 then there exists a root of unity C such that L Q QC D The above theorem says that any abelian extension of Q is contained in a cyclotomic eld We won7t show the whole proof here7 but a rather mini version I will try to convince you in class that in fact it is quite mini indeed Nevertheless7 the results that we show are actually part of the general proof in Rib First7 let7s show a reduction step that uses Galois Theory in an essential way THE REDUCTION STEP If the theorem is true for Abelian extensions having degree a power of a prime7 then it is true for any nite Abelian extension of Q 6 KRONECKER WEBER FOR KIDS Proof Let g GalLQ and let L be a number eld and an Abelian extension of Q of degree n We will show that L is the compo sition of nitely many elds L17 7LS and each Ll has degree a power of a prime To do that we will use the Fundamental Theorem of Finite Abelian Groups By this theorem we have 9 7113 i1 where is a power of a prime not necessarily the same one for all is For any 239 such that 1 S 239 S 5 de ne l Hj i Of course7 all of the s are subgroups of 9 so let L1 be the xed eld of Q Then by the Fundamental Theorem of Galois Theory we have Li 3 Ql lg 3 ill Wit which is a power of a prime Moreover7 any automorphism that xes L1 LS must x all of the Ls7 so we have GalLL1Ls Q 0 thus L L1L5 Now assuming that the theorem is true for all the Ls we may write Li Q 507 where is a primitive root of unity Let C be a primitive root of unity of order equal to the least common multiple of the orders of the Q s Then LL1quot39Ls QQ 177 s QQK And this nishes the proof of the reduction step D We will show an easy special case of the theorem when the degree of the extension and the discriminant are a power of the same odd rational prime Even in this case we will make some assumptions7 which are theorems in their own right As usual we will refer the reader to Mar of Rib for the proofs From now on p will denote an odd prime number of Z CASE 1 If L Q pm7 6L pk7 where mk 2 17 then the theorem holds KRONECKER WEBER FOR KIDS 7 Proof Let R QC where C is a primitive root of unity of order pm We have shown in class that RQ has degree pmp 7 I and GalRQ ZpmHZ and is cyclic We assume that SR is a power of p Let R be the xed eld of the subgroup of GalRQ of order p71 Then we have PU Q pm and also R is a cyclic extension We also see that 633 is a power of p since if q divides 633 then it rami es in R and therefore is must ramify in R but then q SR and so q p Consider the composite eld LR By a problem we had to do on our exam we have LR Q LR R HR Q L L R HR Q so LR Q is a power of a prime Assume that 5ng is a power of the same prime we can show that this is the case by an argument similar to the one for 633 R QK LR R R L J It can be shown that LR Q is a cyclic extension so assume that then the subgroup of GalLR Q GalLR L R is also cyclic But by Galois Theory GalLR L R E GalLL R gtlt GalR L R Thus one of the groups on the right is trivial ie L L R or R L R IfLL R then L QR ifR L R thenR Q L but both have the same degree over Q and thus in either case LgR gQC D Combining this with our reduction step we have shown that MINI KRONECKER WEBER IfL L1 Lt is a nite Abelian Extension of Q and if for 1 S 239 S t we have Ll Q p2 6L pi for some odd primes p and m k 2 1 then there exists a primitive root of unity C such that L Q QC D 8 KRONECKER WEBER FOR KIDS It is rather unfortunate that the condition on the discriminant is almost never true I will show some very chilling examples of this in class Finally we show the following theorem which will expand the elds that we can use in the reduction step even further KRONECKER WEBER FOR QUADRATIC FIELDS If LQ is a number eld of degree 2 then there exists a root of unity C such that L Q QK Proof WLOG assume that L Qd where d is square free From a homework problem we know that LQ is Galois over Q so we can use it in our reduction step Fix the notation n to mean a primitive nth root of unity Since d is square free we have d i2ep1ph where e 0 or I r 2 0 and all the ps are odd primes but then Now we tackle each of the elements we have adjoined separately First xjl 4 Thus xjl E Q 4 C Q 8 Also we have lt s glgt2 22 4 12 2 so 6 Q s Finally we have to deal with the case p p 31 2 Consider the group ZpZX We know that this is a cyclic group so for exactly half of the elements a E ZpZgtlt the equation a 2 has a solution while the other half make such an equation impossible De ne the Legendre Symbol 11 by 11 1 iff a is an element such that the equation a x2 can be solved and 11 71 iff 1 makes the equation impossible to solve It is not dif cult to see that is a group homomorphism from ZpZgtlt onto i1 Since we are starting to learn about representation theory in class lets be ridiculous and observe that is in fact a character since we can identify 1 with the n gtlt 71 identity matrix I for any Z21 Look at the sum in Q p ltgt zEZpZX p Using the fact that the Legendre symbol is a homomorphism we have Z 1 KRONECKER WEBER FOR KIDS 9 But we can write y z t for some t E ZpZX7 yielding T2 Z 620 mt6ZpZX Z 35M1H p mt6ZpZX p p Z 35M1H t p p 16ZpZX t mlt Z Z 9 tEZpZX zEZPZX Observe that if 1 t 31 0 then we have 2530 612 55 54 1 while if1t0then Z 1t111pil m The last thing to notice is that since there are an equal number of elements that have equal to 1 as there are with 71 we have Zia10 0 Returning to the calculation of 7392 this gives ip Thus M E Q p Which immediately gives V16 6 Q p or V15 6 Qj17 p Q Qlt s7 pl Combining everything that we have found out so far we get L Qbg Q Qlt 87 p177 p gt Q QC7 where C is a primitive root of unity of degree m 8101 pT D We have shown an improved version of our mini Kronecker Weber 10 KRONECKER WEBER FOR KIDS KRONECKER WEBER FOR KIDS lf L L1Lt is a nite Abelian Extension of Q and for 1 S 239 S t we have Ll Q 10 FL 10 for some odd primes p and mk 2 1 OR Ll Q 2 then there exists a primitive root of unity C such that L Q QC D It seems most appropriate to nish with an example Tracing back through our proofs we see that we can be certain that the composite eld N Qlt Qlt H g 530 ln class I will use an on line MAGMA a computer algebra system calculator to show that N Q 20 4 REFERENCES The books that were used in preparation for this handout are Mar D Marcus Number Fields Universitext series of Springer Rib P Ribenboim Classical Theory ofAlgebrale Numbers Univer sitext series of Springer Both can be found in the Reed College library D ALEKSANDAR PETROV REED COLLEGE MS 986 3203 SE WOODSTOCK BLVDi PORTLAND OR 97202 E mail address aleks reed edu
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