Showing posts with label 1. Show all posts
Showing posts with label 1. Show all posts
Wednesday, November 19, 2014
Splitting of prime ideals in quadratic extensions of ℚ part 1
Our discussion of algebraic number theory returns by popular demand. Way back last April we presented some generalities on factorization of prime ideals in extension fields. (For explanation of what that means, including other necessary concepts, youll have to review earlier installments of this series, which can be found here.)
If this is all Greek to you, I apologize, but thats unavoidable at this stage of a rather technical subject. You may want to go back to the earliest parts of the series to see how the subject got its start and why it may be interesting.
The discussion in the previous installment probably seems rather dry and abstract, but when we look at simple examples, such as quadratic extensions, why its interesting becomes clearer.
Because of how the ramification indexes and inertial degrees are related, for any prime ideal (p) of ℤ there are only three different possibilities for how the ideal factors in the ring of integers of a quadratic extension:
It turns out that there are simple criteria for each of these cases. But figuring out what the criteria are is tricky.
Recall that in ℚ(√3) we found (13)=(4+√3)⋅(4-√3) and (-11)=(8+5√3)⋅(8-5√3), so both (11) and (13) split completely. Clearly, (3)=(√3)2, so (3) is an example of a prime ideal if ℤ that is ramified. How about an example of a prime ideal that is inert in the extension? This is a little harder for a couple of reasons. (p) will be inert just in case it neither splits nor is ramified, but we dont yet have simple criteria to rule out those cases.
So lets back up a little and look at the details. We found examples where (p) splits in the integers of ℚ(√3) by solving the equation ±p=a2-3b2, because that gave elements a±b√3 whose norm was ±p. Being able to find such elements guaranteed that the prime split. But ℚ(√d) with d=3 is a special case, since here d≡3 (mod 4). In that case, and also if d≡2 (mod 4), the integers of the extension have the form a+b√d with a,b∈ℤ.
If d≡1 (mod 4), integers can also have the form (a+b√d)/2, with a,b∈ℤ, and we might have a factorization like (p) = ((a+b√d)/2)⋅((a-b√d)/2), so we would have also to consider solvability of ±4p=a2-3b2. If we were to look at solvability of the approriate equation, according as to whether or not d≡1 (mod 4), the solvability would be a sufficient condition for (p) to split (or ramify if a=0). Notice that this sufficient condition for (p) to split holds regardless of whether or not ℚ(√d) is a PID.
Now we need to find a convenient necessary condition for (p) to split. Unfortunately, solvability of one simple equation is not a necessary condition in general. It would be, as well see in a minute, if the ring of integers of ℚ(√d) happens to be a PID, as is true when d=3. However, in quadratic extensions where the ring of integers isnt a PID, being unable to solve the applicable equation doesnt guarantee (p) cannot split, because there might be non-principal ideals that are factors of (p).
So lets ignore that problem for a moment and just focus on the case where the ring of integers of ℚ(√d) is a PID. Can we then find a necessary condition on p for (p) to split or ramify, i. e. for (p) to not be a prime ideal of the integers of ℚ(√d)? That is, what must be true about p if (p) splits or ramifies?
If (p) splits or ramifies, then (p)=P1⋅P2 for nontrivial ideals Pi. (The ideals are the same or distinct according as (p) ramifies or splits.) Assuming ℚ(√d) is a PID, then P1 is generated by a+b√d where both a,b∈ℤ, if d≡2 or 3 (mod 4), or else by (a+b√d)/2 with a,b∈ℤ, if d≡1 (mod 4). Likewise, the conjugate ideal P2 is generated by a-b√d or (a-b√d)/2. Since p∈P1⋅P2, by definition of a product of ideals, p is of the form p = ε(a+b√d)(a-b√d) = ε(a2-db2) or p = ε(a+b√d)(a-b√d)/4 = ε(a2-db2)/4 for some integer ε of ℚ(√d).
Recall that the norm of an element of a Galois extension field is the product of all conjugates of the element. So for an element that is also in the base field, the norm (with respect to a quadratic extension, which is always Galois) is the square of the element. Taking norms of both sides of the possible equations, then either p2 = N(&epsilon)(a2-db2)2 or 16p2 = N(&epsilon)(a2-db2)2. For simplicity, consider just the first case. Then N(ε) is a positive integer that has to be 1, p, or p2. If N(ε)≠1 then N(a±b√d) = a2-db2 must be ±1, so a±b√d must be a unit, and both Pi must be non-proper ideals (i. e. equal to the whole ring). Hence N(ε)=1. This will be true also in the other case (when d≡1 (mod 4)), so ±p=a2-db2 or ±4p=a2-db2. Consequently, solvability of the appropriate equation (depending on d mod 4), is a necessary condition for (p) to split or ramify.
So we have a necessary and sufficient condition for (p) to split or ramify in ℚ(√d), in terms of solvability of Diophantine equations, provided Oℚ(√d) is a PID. Since the only other possibility is for (p) to be inert, we also have a necessary and sufficient condition for that.
However, still assuming that Oℚ(√d) is a PID, we can find a further necessary condition for (p) to split or ramify. Take those equations we just found and reduce them modulo p. Then both equations become a2≡db2 (mod p). Since p is prime, ℤ/pℤ is a field. Assume first that b≢0 (mod p). Then b has an inverse in the finite field. So we have d≡(a/b)2 (mod p). In other words, d is a square mod p. This is the additional necessary condition we were looking for on p in order for (p) to split or ramify. Since its a necessary condition, if d is not a square mod p, then (p) must not split or ramify, and thus p is inert. And so, for d to be a non-square mod p is a sufficient condition for p to be inert.
(What if b≡0 (mod p)? Then b=b1p. So ±p = a2 - (b1p)2d or else ±4p = a2 - (b1p)2d. Either way, p∣a, hence p2 divides the right side of either equation, and hence the left side also. But thats not possible unless p=2 – which is always a special case.)
To summarize, then, let p≠2 be prime and d square-free and not 0 or 1. Then the solvability of ±tp=a2-db2 (where t is 4 or 1 according as d≡1 (mod 4) or not), is sufficient for (p) to split or ramify. And if the integers of ℚ(√d) are a PID, then solvability of the appropriate equation provides a necessary and sufficient condition for (p) to split or ramify. Further, in that case, d being a square mod p is necessary for (p) to split or ramify.
Remarkably, d being a square mod p is a necessary and sufficient condition for (p) to split or ramify, even if the integers of ℚ(√d) arent a PID, but thats harder to prove. Since solvability of Diophantine equations is generally not obvious by inspection, its very convenient to have a necessary and sufficient conditions for (p) to split or ramify simply in terms of the properties of d mod p.
In the next installment, which hopefully will not be as long in coming as this one, well show a much cleaner way to state necessary and sufficient conditions for (p) to split, ramify, or remain inert, in the case of any quadratic extension of ℚ, whether or not the ring of integers is a PID. This will be done in terms of what has long been called a "reciprocity law".
However, that will be only the beginning. It turns out that there are far more general kinds of reciprocity laws for many other types of field extensions. Thats what "class field theory" is all about, and why the whole subject is so appealing, once you get the basic ideas.
Tags: algebraic number theory
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If this is all Greek to you, I apologize, but thats unavoidable at this stage of a rather technical subject. You may want to go back to the earliest parts of the series to see how the subject got its start and why it may be interesting.
The discussion in the previous installment probably seems rather dry and abstract, but when we look at simple examples, such as quadratic extensions, why its interesting becomes clearer.
Because of how the ramification indexes and inertial degrees are related, for any prime ideal (p) of ℤ there are only three different possibilities for how the ideal factors in the ring of integers of a quadratic extension:
- (p)=P1⋅P2, so (p) splits completely. (e=f=1, g=2)
- (p)=P is a prime ideal in ℚ(√d), so p is inert. (e=g=1, f=2)
- (p)=P2 where P is prime, and p is ramified. (f=g=1, e=2)
It turns out that there are simple criteria for each of these cases. But figuring out what the criteria are is tricky.
Recall that in ℚ(√3) we found (13)=(4+√3)⋅(4-√3) and (-11)=(8+5√3)⋅(8-5√3), so both (11) and (13) split completely. Clearly, (3)=(√3)2, so (3) is an example of a prime ideal if ℤ that is ramified. How about an example of a prime ideal that is inert in the extension? This is a little harder for a couple of reasons. (p) will be inert just in case it neither splits nor is ramified, but we dont yet have simple criteria to rule out those cases.
So lets back up a little and look at the details. We found examples where (p) splits in the integers of ℚ(√3) by solving the equation ±p=a2-3b2, because that gave elements a±b√3 whose norm was ±p. Being able to find such elements guaranteed that the prime split. But ℚ(√d) with d=3 is a special case, since here d≡3 (mod 4). In that case, and also if d≡2 (mod 4), the integers of the extension have the form a+b√d with a,b∈ℤ.
If d≡1 (mod 4), integers can also have the form (a+b√d)/2, with a,b∈ℤ, and we might have a factorization like (p) = ((a+b√d)/2)⋅((a-b√d)/2), so we would have also to consider solvability of ±4p=a2-3b2. If we were to look at solvability of the approriate equation, according as to whether or not d≡1 (mod 4), the solvability would be a sufficient condition for (p) to split (or ramify if a=0). Notice that this sufficient condition for (p) to split holds regardless of whether or not ℚ(√d) is a PID.
Now we need to find a convenient necessary condition for (p) to split. Unfortunately, solvability of one simple equation is not a necessary condition in general. It would be, as well see in a minute, if the ring of integers of ℚ(√d) happens to be a PID, as is true when d=3. However, in quadratic extensions where the ring of integers isnt a PID, being unable to solve the applicable equation doesnt guarantee (p) cannot split, because there might be non-principal ideals that are factors of (p).
So lets ignore that problem for a moment and just focus on the case where the ring of integers of ℚ(√d) is a PID. Can we then find a necessary condition on p for (p) to split or ramify, i. e. for (p) to not be a prime ideal of the integers of ℚ(√d)? That is, what must be true about p if (p) splits or ramifies?
If (p) splits or ramifies, then (p)=P1⋅P2 for nontrivial ideals Pi. (The ideals are the same or distinct according as (p) ramifies or splits.) Assuming ℚ(√d) is a PID, then P1 is generated by a+b√d where both a,b∈ℤ, if d≡2 or 3 (mod 4), or else by (a+b√d)/2 with a,b∈ℤ, if d≡1 (mod 4). Likewise, the conjugate ideal P2 is generated by a-b√d or (a-b√d)/2. Since p∈P1⋅P2, by definition of a product of ideals, p is of the form p = ε(a+b√d)(a-b√d) = ε(a2-db2) or p = ε(a+b√d)(a-b√d)/4 = ε(a2-db2)/4 for some integer ε of ℚ(√d).
Recall that the norm of an element of a Galois extension field is the product of all conjugates of the element. So for an element that is also in the base field, the norm (with respect to a quadratic extension, which is always Galois) is the square of the element. Taking norms of both sides of the possible equations, then either p2 = N(&epsilon)(a2-db2)2 or 16p2 = N(&epsilon)(a2-db2)2. For simplicity, consider just the first case. Then N(ε) is a positive integer that has to be 1, p, or p2. If N(ε)≠1 then N(a±b√d) = a2-db2 must be ±1, so a±b√d must be a unit, and both Pi must be non-proper ideals (i. e. equal to the whole ring). Hence N(ε)=1. This will be true also in the other case (when d≡1 (mod 4)), so ±p=a2-db2 or ±4p=a2-db2. Consequently, solvability of the appropriate equation (depending on d mod 4), is a necessary condition for (p) to split or ramify.
So we have a necessary and sufficient condition for (p) to split or ramify in ℚ(√d), in terms of solvability of Diophantine equations, provided Oℚ(√d) is a PID. Since the only other possibility is for (p) to be inert, we also have a necessary and sufficient condition for that.
However, still assuming that Oℚ(√d) is a PID, we can find a further necessary condition for (p) to split or ramify. Take those equations we just found and reduce them modulo p. Then both equations become a2≡db2 (mod p). Since p is prime, ℤ/pℤ is a field. Assume first that b≢0 (mod p). Then b has an inverse in the finite field. So we have d≡(a/b)2 (mod p). In other words, d is a square mod p. This is the additional necessary condition we were looking for on p in order for (p) to split or ramify. Since its a necessary condition, if d is not a square mod p, then (p) must not split or ramify, and thus p is inert. And so, for d to be a non-square mod p is a sufficient condition for p to be inert.
(What if b≡0 (mod p)? Then b=b1p. So ±p = a2 - (b1p)2d or else ±4p = a2 - (b1p)2d. Either way, p∣a, hence p2 divides the right side of either equation, and hence the left side also. But thats not possible unless p=2 – which is always a special case.)
To summarize, then, let p≠2 be prime and d square-free and not 0 or 1. Then the solvability of ±tp=a2-db2 (where t is 4 or 1 according as d≡1 (mod 4) or not), is sufficient for (p) to split or ramify. And if the integers of ℚ(√d) are a PID, then solvability of the appropriate equation provides a necessary and sufficient condition for (p) to split or ramify. Further, in that case, d being a square mod p is necessary for (p) to split or ramify.
Remarkably, d being a square mod p is a necessary and sufficient condition for (p) to split or ramify, even if the integers of ℚ(√d) arent a PID, but thats harder to prove. Since solvability of Diophantine equations is generally not obvious by inspection, its very convenient to have a necessary and sufficient conditions for (p) to split or ramify simply in terms of the properties of d mod p.
In the next installment, which hopefully will not be as long in coming as this one, well show a much cleaner way to state necessary and sufficient conditions for (p) to split, ramify, or remain inert, in the case of any quadratic extension of ℚ, whether or not the ring of integers is a PID. This will be done in terms of what has long been called a "reciprocity law".
However, that will be only the beginning. It turns out that there are far more general kinds of reciprocity laws for many other types of field extensions. Thats what "class field theory" is all about, and why the whole subject is so appealing, once you get the basic ideas.
Tags: algebraic number theory
Thursday, August 28, 2014
Selected readings 1 15 10
Interesting reading and news items.
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These items are also bookmarked at my Diigo account.
- Crashing the size barrier
- Over the past 50 years, the most common method of increasing the energy of a particle accelerator has been to increase its size. Yet that tactic is reaching a breaking point. While even higher energies are needed to answer many of sciences most pressing questions—such as the origin of mass and the identity of dark matter—simply scaling up the current technology is becoming prohibitively expensive. Scientists need less costly, more efficient means of accelerating particles to ever-greater energies. [Symmetry, 10/1/09]
- Are Black Holes the Architects of the Universe?
- Black holes are finally winning some respect. After long regarding them as agents of destruction or dismissing them as mere by-products of galaxies and stars, scientists are recalibrating their thinking. Now it seems that black holes debuted in a constructive role and appeared unexpectedly soon after the Big Bang. “Several years ago, nobody imagined that there were such monsters in the early universe,” says Penn State astrophysicist Yuexing Li. “Now we see that black holes were essential in creating the universe’s modern structure.” [Discover, 1/4/10]
- Genome advances promise personalized medical treatment
- Six years after scientists finished decoding the human genome -- the genetic instruction book for life -- theyre starting to take their new knowledge from the research laboratory to the doctors office and the patients bedside. ... Researchers are seeking ways to tailor treatments to individuals -- they call it "personalized medicine" -- in order to improve patient outcomes and to lower costs in the overburdened U.S. health care system. [Physorg.com, 11/18/09]
- Hunting for Planets in the Dark
- In Europe, the Euclid mission is a proposed space telescope for characterizing dark energy, but some believe that it might be more attractive to funding agencies if it included an exoplanet survey. [Physorg.com, 11/19/09]
- Dark Energy Search Could Aid Planet Hunters
- The search for dark energy might help in the search for life in the universe. Thats because planet hunting through a technique called microlensing requires a similar sort of instrument as a dark energy mission. [Space.com, 11/19/09]
- Recipes for planet formation
- Observations of extrasolar planets are shaping our ideas about how planetary systems form and evolve. Michael R Meyer describes whats cooking elsewhere in our galaxy – and beyond. [Physicsworld.com, 11/2/09]
- The Americanization of Mental Illness
- We have for many years been busily engaged in a grand project of Americanizing the world’s understanding of mental health and illness. We may indeed be far along in homogenizing the way the world goes mad. This unnerving possibility springs from recent research by a loose group of anthropologists and cross-cultural psychiatrists. [New York Times, 1/8/10]
- What Life Leaves Behind
- The search for life beyond our pale blue dot is fraught with dashed hopes. Will the chemical and mineral fingerprints of Earthly organisms apply on other worlds? [Seed, 11/9/09]
- 3 Questions: Sara Seager on searching for Earth-like planets
- MIT planetary scientist Sara Seager has been studying exoplanets — planets circling stars other than the sun — for many years. The first such planet was discovered just 15 years ago, and now more than 400 others are known. This week, a paper co-authored by Seager and NASA scientist Drake Deming in the journal Nature reviews what we know about exoplanets so far, what we can expect to learn about them in the next decade or so, and the chances for finding a twin of our own planet. She has also just published an online book to answer questions about exoplanets and the lessons they hold. [Physorg.com, 11/23/09]
- Quest for the Holy Grail: Sara Seager Seeks to Complete a Revolution
- Sara Seager is fascinated by stories of explorers visiting uncharted places. From her groundbreaking work on the detection of exoplanet atmospheres to her innovative theories about life on other worlds, Seager has been a pioneer in the vast and unknown world of exoplanets. Now, like an astronomical Indiana Jones, shes on a quest after the fields holy grail - another Earth-like planet. [NASA, 10/6/08]
- Two-qubit quantum system used to model the hydrogen molecule
- Even though quantum computers are still in their crawling phase, computer scientists continue to push their limits. Recently, a group of scientists used a two-qubit quantum system to model the energies of a hydrogen molecule and found that using an iterative algorithm to calculate each digit of the phase shift gave very accurate results. Their system, while not directly extensible, has the potential to help map the energies of more complex molecules and could result in significant time and power savings compared to classical computers. [Arstechnica.com, 1/13/10]
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