# = 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2

Size: px
Start display at page:

Download "= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2"

Transcription

1 8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose that b 2 0 a (mod p). We claim that then x 2 a (mod p n ) has a solution x for all n N. Assume that we have a solution x of x 2 a (mod p n ) for some n 1. Then x is coprime to p, so that we can find x 1 1 (x + a/x) (mod 2 p2n ). (This is the standard Newton-Raphson iterative method x 1 = x f(x)/f (x) for solving f(x) = 0, applied to the polynomial f(x) = x 2 a, but (mod p 2n ) instead of in R or C.) Then and x 1 x = 1 2 ( x a x ) = 1 2x ( x 2 a ) 0 (mod p n ), x 2 1 a = 1 ) (x 2 + 2a + a2 a 4 x 2 = 1 ( x a ) 2 4 x = 1 a) 2 4x 2(x2 0 (mod p 2n ) Thus, starting with x 0 such that x 2 0 a (mod p 20 ), we get successively x 1 with x 2 1 a (mod p 21 ), x 2 with x 2 2 a (mod p22 ),..., x k with x 2 k a (mod p2k ),..., with x k+1 x k (mod p 2k ). So, writing the x i in base p, we obtain x 0 = b 0 x 1 = b 0 + b 1 p say, specified (mod p 2 ) x 2 = b 0 + b 1 p + b 2 p 2 + b 3 p 3 say, specified (mod p 4 ) x 3 = b 0 + b 1 p + b 2 p 2 + b 3 p 3 + b 4 p 4 + b 5 p 5 + b 6 p 6 + b 7 p 7 say, specified (mod p 8 ), a p and so on. So, in any sense, is x = i=1 b ip i a root of x 2 a (mod p )? It turns out that, yes, it is: x is a root of x 2 = a in the field Q p of p-adic numbers Valuations. In order to define the fields Q p of p-adic numbers for primes p, we first need to discuss valuations. A valuation on a field F is a map from F to the nonnegative real numbers satisfying For each x F x = 0 iff x = 0; (ZERo) For each x, y F xy = x y ; (HOMomorphism) For each x, y F x + y x + y. (TRIangle) 26

2 27 If in addition For each x, y F x + y max( x, y ), (MAXimum) then is called a nonarchimedean valuation. A valuation that is not nonarchimedean, i.e., for which there exist x, y F such that x + y > max( x, y ), is called archimedean. For instance the standard absolute value on R is archimedean because 2 = 2 = > max( 1, 1 ) = 1. Note that MAX is stronger than TRI in the sense that if MAX is true than TRI is certainly true. So to show that a valuation is nonarchimedean we only need to check that ZER, HOM and MAX hold. Proposition 8.1. For any valuation on a field F we have 1 = 1 = 1 and for n N (defined as the sum of n copies of the identity of F) we have n = n and 1/n = 1/ n. Further, for n, m N we have n/m = n / m. Proof. We have 1 = 1 2 = 1 2, using HOM, so that 1 = 0 or 1. But 1 0 by ZER, so 1 = 1. Also 1 = 1 = ( 1) 2 = 1 2 by HOM, so that 1 = 1 since 1 > 0. Further, n = ( 1)n = 1 n = 1 n = n, and from n (1/n) = 1 we get n 1/n = 1 = 1, so that 1/n = 1/ n. Finally, from n/m = n (1/m) we get n/m = n 1/m = n / m Nonarchimedean valuations. From now on we restrict our attention to nonarchimedean valuations. Proposition 8.2 (Principle of Domination). Suppose that we have a nonarchimedean valuation on a field F, and that x, y F with x y. Then Note the equal sign in this statement! x + y = max( x, y ). Proof. Put s = x + y, and assume w.l.g. that x < y. Then s max( x, y ) = y, while y = s x max( s, x ) = max( s, x ) = s, since otherwise we d have y x. Hence s = y = max( x, y ). Corollary 8.3. Suppose that x 1,...,x n F, with nonarchimedean. Then with equality if x 1 > max( x 2,..., x n ). x ,+x n max( x 1,..., x n ), Proof. Use induction, with the help of MAX, for the inequality. For the equality, put x 1 = y and x x n = x in the Principle of Domination. Corollary 8.4. For nonarchimedean and n Z we have n 1. Proof. Apply the Corollary above with all x i = 1. Then use n = n.

3 28 Lemma 8.5. If is a nonarchimedean valuation on F, then so is α for any α > 0. Proof. It s easily checked that ZER, HOM and MAX still hold when the valuation we start with is taken to the α-th power. [ The same does not apply to TRI we need 0 < α 1 for TRI to still always hold.] 8.4. Nonarchimedean valuations on Q. Corollary 8.6. If is a nonarchimedean valuation on Q with n = 1 for all n N then is trivial, i.e., x = 0 if x = 0 while x = 1 if x 0. Proof. We then have x = 0 by ZER, while n/m = n / m = 1/1 = 1. We ll ignore trivial valuations from now on. Proposition 8.7. If is a nonarchimedean valuation on Q with n < 1 for some n N, then there is a prime p such that {n N : n < 1} = {n N : p divides n}. Proof. Take the smallest positive integer n 1 such that n 1 < 1. We know that n 1 > 1. If n 1 is composite, say n 1 = n 2 n 3 with 1 < n 2, n 3 < n 1, then, by the minimality of n 1, we have n 2 = n 3 = 1, so that n 1 = n 2 n 3 = 1 1 = 1 by HOM, a contradiction. Hence n 1 is prime, = p say. Then for any n with n < 1 we can, by the division algorithm, write n = qp + r where 0 r < p. But then r = n qp max( n, qp ) = max( n, 1 q p ) < 1, as 1 = 1, q 1 and p < 1. By the minimality of p we must have r = 0, so that p n. Next, we show that there is indeed a valuation on Q corresponding to each prime p. We define p by 0 = 0, p p = 1/p and n = 1 for n Z and coprime to p, and p k n/m p = p k for n and m coprime to p. We call this the p-adic valuation on Q. Proposition 8.8. The p-adic valuation on Q is indeed a valuation. Proof. The definition of p ensures that ZER and HOM hold. It remains only to check that MAX holds. Let x = p k n/m and y = p k n /m,where n, m, n m are all coprime to p. Suppose w.l.g. that k k. Then x p = p k p n p / m p = p k as n p = m p = 1 and p p = 1/p. Similarly y p = p k x p. Hence x + y p = p k (nm + p k k n m) mm = p k nm + p k k n m p p k = max( x p mm p, y p ). p as m p = m p = 1 and nm + p k k n m p 1, since nm + p k k n m Z. [Note that the choice of p p = 1/p is not particularly important, as by replacing p by its α-th power as in Lemma 8.5 we can make p p equal any number we like in the interval (0, 1). But we do need to fix on a definite value!]

4 8.5. The p-adic completion Q p of Q. We first recall how to construct the real field R from Q, using Cauchy sequences. Take the ordinary absolute value on Q, and define a Cauchy sequence to be a sequence (a n ) = a 1, a 2,...,a n,... of rational numbers with the property that for each ε > 0 there is an N > 0 such that a n a n < ε for all n, n > N. We define an equivalence relation on these Cauchy sequences by saying that two such sequences (a n ) and (b n ) are equivalent if the interlaced sequence a 1, b 1, a 2, b 2,...,a n, b n,... is also a Cauchy sequence. [Essentially, this means that the sequences tend to the same limit, but as we haven t yet constructed R, where (in general) the limit lies, we can t say that.] Having checked that this is indeed an equivalence relation on these Cauchy sequences, we define R to be the set of all equivalence classes of such Cauchy sequences. We represent each equivalence class by a convenient equivalence class representative; one way to do this is by the standard decimal expansion. So, the class π will be represented by the Cauchy sequence 3, 3.1, 3.14, 3.141, , ,..., which we write as Further, we can make R into a field by defining the sum and product of two Cauchy sequences in the obvious way, and also the reciprocal of a sequence, provided the sequence doesn t tend to 0. [The general unique decimal representation of a real number a is a = ±10 k (d 0 + d d d n 10 n +...), where k Z, and the digits d i are in {0, 1, 2,..., 9}, with d 0 0. Also, it is forbidden that the d i s are all = 9 from some point on, as otherwise we get non-unique representations, e.g., 1 = 10 0 ( ) = 10 1 ( ).] We do the same kind of construction to define the p-adic completion Q p of Q, except that we replace the ordinary absolute value by p in the method to obtain p-cauchy sequences. To see what we should take as the equivalence class representatives, we need the following result. Lemma 8.9. Any rational number m/n with m/n p = 1 can be p-adically approximated arbitrarily closely by a positive integer. That is, for any k N there is an N N such that m/n N p p k. Proof. We can assume that n p = 1 and m p 1. We simply take N = mn, where nn 1 (mod p k ). Then the numerator of m/n N is an integer that is divisible by p k. An immediate consequence of this result is that any rational number (i.e., dropping the m/n p = 1 condition) can be approximated arbitrarily closely by a positive integer times a power of p. Thus one can show that any p-cauchy sequence is equivalent to one containing only those kind of numbers. We write the positive integer N in base p, so that p k N = p k (a 0 +a 1 p+a 2 p 2 + +a r p r ) say, where the a i are base-p digits {0, 1, 2,..., p 1}, and where we can clearly assume that a 0 0 (as otherwise we could increase k by 1). We define Q p, the p-adic numbers, to be the set of all equivalence classes of p-cauchy sequences of elements of Q. Then we have the following. 29

6 Negation. If a = p k ( i=0 a ip i ), then a = p k ((p a 0 ) + ) (p 1 a i )p i, as can be checked by adding a to a (and getting 0!). Note that from all a i {0, 1, 2,..., p 1} and a 0 0 we have that the same applies to the digits of a Reciprocals. If a = p k ( i=0 a ip i ), then i=1 1 a = p k (a 0 + a 1 p + + a i pi +...) say, where for any i the first i digits a 0, a 1,..., a i can be calculated as follows: Putting a 0 +a 1 p+ +a i p i = N, calculate N N with N < p i+1 such that NN 1 (mod p i+1 ). Then writing N in base p as N = a 0 + a 1 p + + a i pi gives a 0, a 1,...,a i Addition and multiplication. If a = p k ( i=0 a ip i ) and a = p k ( i=0 a i pi ) (same k) then a+a = p k ((a 0 +a 0)+(a 1 +a 1)p+ +(a i +a i)p i +...), where then carrying needs to be performed to get the digits of a + a into {0, 1, 2,..., p 1}. If a = p k ( i=0 a i pi ) with k < k then we can pad the expansion of a with initial zeros so that we can again assume that k = k, at the expense of no longer having a 0 nonzero. Then addition can be done as above. Multiplication is similar: multiplying a = p k ( i=0 a ip i ) by a = p k ( i=0 a i pi ) gives i a a = p k+k (a 0 a 0 + (a 1a 0 + a 0a 1 )p + + ( a j a i j )pi +...), where then this expression can be put into standard form by carrying Expressing rationals as p-adic numbers. Any nonzero rational can clearly be written as ±p k m/n, where m, n are positive integers coprime to p (and to each other), and k Z. It s clearly enough to express ±m/n as a p-adic number a 0 + a 1 p +..., as then ±p k m/n = p k (a 0 + a 1 p +...) Representating m/n, where 0 < m < n. We have the following result. Proposition Put e = ϕ(n). Suppose that m and n are coprime to p, with 0 < m < n, and that the integer in base p. Then m pe 1 n is written as j=0 d 0 + d 1 p + + d e 1 p e 1 m n = d 0 + d 1 p + + d e 1 p e 1 + d 0 p e + d 1 p e d e 1 p 2e 1 + d 0 p 2e + d 1 p 2e

7 32 Proof. We know that pe 1 n is an integer, by Euler s Theorem. Hence m 1 n = mpe n 1 p = (d e 0 + d 1 p + + d e 1 p e 1 )(1 + p e + p 2e +...), which gives the result. In the above proof, we needed m < n so that m pe 1 n d 0 + d 1 p + + d e 1 p e 1. < p e, and so had a representation The case m/n, where 0 < m < n. For this case, first write m/n = u/(1 p e ), where, as above, u = m pe 1. Then n m n = u 1 p = 1 + pe 1 u = 1 + u e 1 p e 1 p e, where u = p e 1 u and 0 u < p e. Thus we just have to add 1 to the repeating p-adic integer u + u p e + u p 2e Example What is 1/7 in Q 5? From (mod 7) (Fermat), and (5 6 1)/7 = 2232, we have 1 7 = = = (21423)( ) = Hence 1 = , 7 which is a way of writing Taking square roots in Q p The case of p odd. First consider a p-adic unit a = a 0 + a 1 p + a 2 p 2 + Z p, where p is odd. Which such a have a square root in Q p? Well, if a = b 2, where b = b 0 + b 1 p + b 2 p 2 + Z p, then, working modulo p we see that a 0 b 2 0 (mod p), so that a 0 must be a quadratic residue (mod p). In this case the method in Section 8.1 will construct b. Note that if at any stage you are trying to construct b (mod n) then you only need to specify a (mod n), so that you can always work with rational integers rather than with p-adic integers. On the other hand, if a 0 is a quadratic nonresidue, then a has no square root in Q p. Example. Computing 6 in Q 5. While the algorithm given in the introduction to this chapter is a good way to compute square roots by computer, it is not easy to use

8 by hand. Here is a simple way to compute square roots digit-by-digit, by hand: Write 6 = b0 + b b Then, squaring and working mod 5, we have b (mod 5), so that b 0 = 1 or 4. Take b 0 = 1 (4 will give the other square root, which is minus the one we re computing.) Next, working mod 5 2, we have 6 (1 + b 1 5) 2 (mod 5 2 ) b 1 (mod 5 2 ) 1 2b 1 (mod 5), giving b 1 = 3. Doing the same thing mod 5 3 we have 6 ( b ) 2 (mod 5 3 ) b (mod 5 3 ) b (mod 5 3 ) 0 32b 2 (mod 5), giving b 2 = 0. Continuing mod 5 4, we get b 3 = 4, so that 6 = Next, consider a general p-adic number a = p k (a 0 +a 1 p+...). If a = b 2, then a p = b 2 p, so that b p = a p 1/2 = p k/2. But valuations of elements of Q p are integer powers of p, so that if k is odd then b Q p. But if k is even, there is no problem, and a will have a square root b = p k/2 (b 0 + b 1 p +...) Q p iff a 0 is a quadratic residue (mod p) The case of p even. Consider a 2-adic unit a = 1 + a a Z 2. If a = b 2, where b = b 0 + b b Z 2, working modulo 8, we have b 2 1 (mod 8), so that we must have a 1 (mod 8), giving a 1 = a 2 = 0. When this holds, the construction of Section 8.1 will again construct b. On the other hand, if a 1 (mod 8), then a has no square root in Q 2. For a general 2-adic number a = 2 k (1 + a a ), we see that, similarly to the case of p odd, a will have a square root in Q 2 iff k is even and a 1 = a 2 = The Local-Global Principle. The fields Q p (p prime) and R, and their finite extensions, are examples of local fields. These are complete fields, because they contain all their limit points. On the other hand, Q and its finite extensions are called number fields and are examples of global fields. [Other examples of global and local fields are the fields F(x) of rational functions over a finite field F (global) and their completions with respect to the valuations on them (local).] One associates to a global field the local fields obtained by taking the completions of the field with respect to each valuation on that field. Suppose that you are interested in whether an equation f(x, y) = 0 has a solution x, y in rational numbers. Clearly, if the equation has no solution in R, or in some Q p, then, since these fields contain Q, the equation has no solution on Q either. 33

9 34 For example, the equation x 2 + y 2 = 1 has no solution in Q because it has no solution in R. The equation x 2 + 3y 2 = 2 has no solution in Q because it has no solution in Q 3, because 2 is a quadratic nonresidue of 3. The Local-Global (or Hasse-Minkowski) Principle is said to hold for a class of equations (over Q, say) if, whenever an equation in that class has a solution in each of its completions, it has a solution in Q. This principle holds, in particular, for quadratic forms. Thus for such forms in three variables, we have the following result. Theorem Let a, b, c be nonzero integers, squarefree, pairwise coprime and not all of the same sign. Then the equation ax 2 + by 2 + cz 2 = 0 (1) has a nonzero solution (x, y, z) Z 3 iff bc is a quadratic residue of a; i.e. the equation x 2 bc (mod a) has a solution x; ca is a quadratic residue of b; ab is a quadratic residue of c. (Won t prove.) The first of these conditions is necessary and sufficient for (1) to have a solution in Q p for each odd prime dividing a. Similarly for the other two conditions. The condition that a, b, c are not all of the same sign is clearly necessary and sufficent that (1) has a solution in R. But what about a condition for a solution in Q 2? Hilbert symbols. It turns out that we don t need to consider solutions in Q 2, because if a quadratic form has no solution in Q then it has no solution in a positive, even number (so, at least 2!) of its completions. Hence, if we ve checked that it has a solution in all its completions except one, it must in fact have a solution in all its completions, and so have a solution in Q. This is best illustrated by using Hilbert symbols and Hilbert s Reciprocity Law. For a, b Q the Hilbert symbol (a, b) p, where p is a prime or, and Q = R, is defined by (a, b) p = { 1 if ax 2 + by 2 = z 2 has a nonzero solution in Q p ; 1 otherwise. Hilbert s Reciprocity Law says that p (a, b) p = 1. (Won t prove; it is, however, essentially equivalent to the Law of Quadratic Reciprocity.) Hence, a finite, even number of (a, b) p (p a prime or ) are equal to Nonisomorphism of Q p and Q q. When one writes rational numbers to any (integer) base b 2, and then forms the completion with respect to the usual absolute value, one obtains the real numbers R, (though maybe written in base b). Thus the field obtained (R) is independent of b. Furthermore, b needn t be prime. However, when completing Q (in whatever base) with respect to the p-adic valuation to obtain Q p, the field obtained does depend on p, as one might expect, since a different valuation is being used for each p. One can, however, prove this directly:

10 Theorem Take p and q to be two distinct primes. Then Q p and Q q are not isomorphic. Proof. We can assume that p is odd. Suppose first that q is also odd. Let n be a quadratic nonresidue (mod q). Then using the Chinese Remainder ( ) ( Theorem ) we( can ) find ( k, ) l N with 1+kp = n+lq. Hence, for a = 1+kp we have = = 1 while = = 1. Hence, by the results of Subsection 8.8 we see that a Q p but a Q q. Thus, if there were an isomorphism φ : Q p Q q then we d have φ( a) 2 = φ( a 2 ) = φ(a) = φ( ) = a, so that φ( a) would be a square root of a in Q q, a contradiction. Similarly, if q = 2 then we can find a = 1 + kp = 3 + 4l, so that a Q p again, but a Q2. so the same argument applies. Note that for any integer b 2 one can, in fact, define the ring of b-adic numbers, which consists of numbers p k (a 0 + a 1 b + a 2 b a i b i +...), where k Z and all a i {0, 1, 2,..., b 1}. However, if b is composite, this ring has nonzero zero divisors (nonzero numbers a, a such that aa = 0), so is not a field. See problem sheet 5 for the example b = 6. a p 1 p a q n q 35

### MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 57 5. p-adic Numbers 5.1. Motivating examples. We all know that 2 is irrational, so that 2 is not a square in the rational field Q, but that we can

### 1 Adeles over Q. 1.1 Absolute values

1 Adeles over Q 1.1 Absolute values Definition 1.1.1 (Absolute value) An absolute value on a field F is a nonnegative real valued function on F which satisfies the conditions: (i) x = 0 if and only if

### Commutative Rings and Fields

Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two

### Chapter 8. P-adic numbers. 8.1 Absolute values

Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.

### Material covered: Class numbers of quadratic fields, Valuations, Completions of fields.

ALGEBRAIC NUMBER THEORY LECTURE 6 NOTES Material covered: Class numbers of quadratic fields, Valuations, Completions of fields. 1. Ideal class groups of quadratic fields These are the ideal class groups

### Introduction to Arithmetic Geometry Fall 2013 Lecture #5 09/19/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #5 09/19/2013 5.1 The field of p-adic numbers Definition 5.1. The field of p-adic numbers Q p is the fraction field of Z p. As a fraction field,

### Summary Slides for MATH 342 June 25, 2018

Summary Slides for MATH 342 June 25, 2018 Summary slides based on Elementary Number Theory and its applications by Kenneth Rosen and The Theory of Numbers by Ivan Niven, Herbert Zuckerman, and Hugh Montgomery.

### Course 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography

Course 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2006 Contents 9 Introduction to Number Theory and Cryptography 1 9.1 Subgroups

### Part II. Number Theory. Year

Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler

### Course MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography

Course MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2000 2013 Contents 9 Introduction to Number Theory 63 9.1 Subgroups

### Local Fields. Chapter Absolute Values and Discrete Valuations Definitions and Comments

Chapter 9 Local Fields The definition of global field varies in the literature, but all definitions include our primary source of examples, number fields. The other fields that are of interest in algebraic

### Lecture notes: Algorithms for integers, polynomials (Thorsten Theobald)

Lecture notes: Algorithms for integers, polynomials (Thorsten Theobald) 1 Euclid s Algorithm Euclid s Algorithm for computing the greatest common divisor belongs to the oldest known computing procedures

### The group (Z/nZ) February 17, In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer.

The group (Z/nZ) February 17, 2016 1 Introduction In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer. If we factor n = p e 1 1 pe, where the p i s are distinct

### a = a i 2 i a = All such series are automatically convergent with respect to the standard norm, but note that this representation is not unique: i<0

p-adic Numbers K. Sutner v0.4 1 Modular Arithmetic rings integral domains integers gcd, extended Euclidean algorithm factorization modular numbers add Lemma 1.1 (Chinese Remainder Theorem) Let a b. Then

### Mathematics for Cryptography

Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, 2016 1 Groups and Modular Arithmetic 1.1

### SOLUTIONS TO PROBLEM SET 1. Section = 2 3, 1. n n + 1. k(k + 1) k=1 k(k + 1) + 1 (n + 1)(n + 2) n + 2,

SOLUTIONS TO PROBLEM SET 1 Section 1.3 Exercise 4. We see that 1 1 2 = 1 2, 1 1 2 + 1 2 3 = 2 3, 1 1 2 + 1 2 3 + 1 3 4 = 3 4, and is reasonable to conjecture n k=1 We will prove this formula by induction.

### The p-adic Numbers. Akhil Mathew

The p-adic Numbers Akhil Mathew ABSTRACT These are notes for the presentation I am giving today, which itself is intended to conclude the independent study on algebraic number theory I took with Professor

### Introduction to Number Theory

INTRODUCTION Definition: Natural Numbers, Integers Natural numbers: N={0,1,, }. Integers: Z={0,±1,±, }. Definition: Divisor If a Z can be writeen as a=bc where b, c Z, then we say a is divisible by b or,

### The primitive root theorem

The primitive root theorem Mar Steinberger First recall that if R is a ring, then a R is a unit if there exists b R with ab = ba = 1. The collection of all units in R is denoted R and forms a group under

### 1 Take-home exam and final exam study guide

Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number

### 2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.

2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say

### Rational Points on Conics, and Local-Global Relations in Number Theory

Rational Points on Conics, and Local-Global Relations in Number Theory Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu http://www.math.purdue.edu/ lipman November 26, 2007

### LEGENDRE S THEOREM, LEGRANGE S DESCENT

LEGENDRE S THEOREM, LEGRANGE S DESCENT SUPPLEMENT FOR MATH 370: NUMBER THEORY Abstract. Legendre gave simple necessary and sufficient conditions for the solvablility of the diophantine equation ax 2 +

### Chapter 5. Modular arithmetic. 5.1 The modular ring

Chapter 5 Modular arithmetic 5.1 The modular ring Definition 5.1. Suppose n N and x, y Z. Then we say that x, y are equivalent modulo n, and we write x y mod n if n x y. It is evident that equivalence

### MATH 117 LECTURE NOTES

MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set

### MATH 361: NUMBER THEORY FOURTH LECTURE

MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the

### Corollary 4.2 (Pepin s Test, 1877). Let F k = 2 2k + 1, the kth Fermat number, where k 1. Then F k is prime iff 3 F k 1

4. Primality testing 4.1. Introduction. Factorisation is concerned with the problem of developing efficient algorithms to express a given positive integer n > 1 as a product of powers of distinct primes.

### OSTROWSKI S THEOREM FOR Q(i)

OSTROWSKI S THEOREM FOR Q(i) KEITH CONRAD We will extend Ostrowki s theorem from Q to the quadratic field Q(i). On Q, every nonarchimedean absolute value is equivalent to the p-adic absolute value for

### Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively

6 Prime Numbers Part VI of PJE 6.1 Fundamental Results Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively D (p) = { p 1 1 p}. Otherwise

### Winter Camp 2009 Number Theory Tips and Tricks

Winter Camp 2009 Number Theory Tips and Tricks David Arthur darthur@gmail.com 1 Introduction This handout is about some of the key techniques for solving number theory problems, especially Diophantine

### CONSTRUCTION OF THE REAL NUMBERS.

CONSTRUCTION OF THE REAL NUMBERS. IAN KIMING 1. Motivation. It will not come as a big surprise to anyone when I say that we need the real numbers in mathematics. More to the point, we need to be able to

### Notes on Systems of Linear Congruences

MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the

### The p-adic Numbers. Akhil Mathew. 4 May Math 155, Professor Alan Candiotti

The p-adic Numbers Akhil Mathew Math 155, Professor Alan Candiotti 4 May 2009 Akhil Mathew (Math 155, Professor Alan Candiotti) The p-adic Numbers 4 May 2009 1 / 17 The standard absolute value on R: A

### The p-adic numbers. Given a prime p, we define a valuation on the rationals by

The p-adic numbers There are quite a few reasons to be interested in the p-adic numbers Q p. They are useful for solving diophantine equations, using tools like Hensel s lemma and the Hasse principle,

### 1. multiplication is commutative and associative;

Chapter 4 The Arithmetic of Z In this chapter, we start by introducing the concept of congruences; these are used in our proof (going back to Gauss 1 ) that every integer has a unique prime factorization.

### Solutions, Project on p-adic Numbers, Real Analysis I, Fall, 2010.

Solutions, Project on p-adic Numbers, Real Analysis I, Fall, 2010. (24) The p-adic numbers. Let p {2, 3, 5, 7, 11,... } be a prime number. (a) For x Q, define { 0 for x = 0, x p = p n for x = p n (a/b),

### Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

### MATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions

MATH 11/CSCI 11, Discrete Structures I Winter 007 Toby Kenney Homework Sheet 5 Hints & Model Solutions Sheet 4 5 Define the repeat of a positive integer as the number obtained by writing it twice in a

### Number Theory, Algebra and Analysis. William Yslas Vélez Department of Mathematics University of Arizona

Number Theory, Algebra and Analysis William Yslas Vélez Department of Mathematics University of Arizona O F denotes the ring of integers in the field F, it mimics Z in Q How do primes factor as you consider

### Math 109 HW 9 Solutions

Math 109 HW 9 Solutions Problems IV 18. Solve the linear diophantine equation 6m + 10n + 15p = 1 Solution: Let y = 10n + 15p. Since (10, 15) is 5, we must have that y = 5x for some integer x, and (as we

### THE P-ADIC NUMBERS AND FINITE FIELD EXTENSIONS OF Q p

THE P-ADIC NUMBERS AND FINITE FIELD EXTENSIONS OF Q p EVAN TURNER Abstract. This paper will focus on the p-adic numbers and their properties. First, we will examine the p-adic norm and look at some of

### HASSE-MINKOWSKI THEOREM

HASSE-MINKOWSKI THEOREM KIM, SUNGJIN 1. Introduction In rough terms, a local-global principle is a statement that asserts that a certain property is true globally if and only if it is true everywhere locally.

### ALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers

ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some

### 4 Powers of an Element; Cyclic Groups

4 Powers of an Element; Cyclic Groups Notation When considering an abstract group (G, ), we will often simplify notation as follows x y will be expressed as xy (x y) z will be expressed as xyz x (y z)

### NOTES ON FINITE FIELDS

NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining

### An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p.

Chapter 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p. If n > 1

### Math 118: Advanced Number Theory. Samit Dasgupta and Gary Kirby

Math 8: Advanced Number Theory Samit Dasgupta and Gary Kirby April, 05 Contents Basics of Number Theory. The Fundamental Theorem of Arithmetic......................... The Euclidean Algorithm and Unique

### LECTURE 2. Hilbert Symbols

LECTURE 2 Hilbert Symbols Let be a local field over Q p (though any local field suffices) with char() 2. Note that this includes fields over Q 2, since it is the characteristic of the field, and not the

### 3 The fundamentals: Algorithms, the integers, and matrices

3 The fundamentals: Algorithms, the integers, and matrices 3.4 The integers and division This section introduces the basics of number theory number theory is the part of mathematics involving integers

### P-adic numbers. Rich Schwartz. October 24, 2014

P-adic numbers Rich Schwartz October 24, 2014 1 The Arithmetic of Remainders In class we have talked a fair amount about doing arithmetic with remainders and now I m going to explain what it means in a

### CHAPTER 6. Prime Numbers. Definition and Fundamental Results

CHAPTER 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results 6.1. Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and the only positive divisors of p are 1 and p. If n

### Number Theory Homework.

Number Theory Homewor. 1. The Theorems of Fermat, Euler, and Wilson. 1.1. Fermat s Theorem. The following is a special case of a result we have seen earlier, but as it will come up several times in this

### 2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?

Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative

### Basic elements of number theory

Cryptography Basic elements of number theory Marius Zimand 1 Divisibility, prime numbers By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a

### A Guide to Arithmetic

A Guide to Arithmetic Robin Chapman August 5, 1994 These notes give a very brief resumé of my number theory course. Proofs and examples are omitted. Any suggestions for improvements will be gratefully

### Basic elements of number theory

Cryptography Basic elements of number theory Marius Zimand By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a k for some integer k. Notation

### MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM

MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM Basic Questions 1. Compute the factor group Z 3 Z 9 / (1, 6). The subgroup generated by (1, 6) is

### Mathematical Induction

Mathematical Induction Representation of integers Mathematical Induction Reading (Epp s textbook) 5.1 5.3 1 Representations of Integers Let b be a positive integer greater than 1. Then if n is a positive

### Part IA Numbers and Sets

Part IA Numbers and Sets Definitions Based on lectures by A. G. Thomason Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly)

### NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:

NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two operations defined on them, addition and multiplication,

### All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points.

Math 152, Problem Set 2 solutions (2018-01-24) All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points. 1. Let us look at the following equation: x 5 1

### Public-key Cryptography: Theory and Practice

Public-key Cryptography Theory and Practice Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Chapter 2: Mathematical Concepts Divisibility Congruence Quadratic Residues

### The Chinese Remainder Theorem

Chapter 5 The Chinese Remainder Theorem 5.1 Coprime moduli Theorem 5.1. Suppose m, n N, and gcd(m, n) = 1. Given any remainders r mod m and s mod n we can find N such that N r mod m and N s mod n. Moreover,

### ECEN 5022 Cryptography

Elementary Algebra and Number Theory University of Colorado Spring 2008 Divisibility, Primes Definition. N denotes the set {1, 2, 3,...} of natural numbers and Z denotes the set of integers {..., 2, 1,

### Chapter 5: The Integers

c Dr Oksana Shatalov, Fall 2014 1 Chapter 5: The Integers 5.1: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition

### In N we can do addition, but in order to do subtraction we need to extend N to the integers

Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend

### φ(xy) = (xy) n = x n y n = φ(x)φ(y)

Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =

### 7. Prime Numbers Part VI of PJE

7. Prime Numbers Part VI of PJE 7.1 Definition (p.277) A positive integer n is prime when n > 1 and the only divisors are ±1 and +n. That is D (n) = { n 1 1 n}. Otherwise n > 1 is said to be composite.

### SOME AMAZING PROPERTIES OF THE FUNCTION f(x) = x 2 * David M. Goldschmidt University of California, Berkeley U.S.A.

SOME AMAZING PROPERTIES OF THE FUNCTION f(x) = x 2 * David M. Goldschmidt University of California, Berkeley U.S.A. 1. Introduction Today we are going to have a look at one of the simplest functions in

### CYCLICITY OF (Z/(p))

CYCLICITY OF (Z/(p)) KEITH CONRAD 1. Introduction For each prime p, the group (Z/(p)) is cyclic. We will give seven proofs of this fundamental result. A common feature of the proofs that (Z/(p)) is cyclic

### Basic Algorithms in Number Theory

Basic Algorithms in Number Theory Algorithmic Complexity... 1 Basic Algorithms in Number Theory Francesco Pappalardi Discrete Logs, Modular Square Roots & Euclidean Algorithm. July 20 th 2010 Basic Algorithms

### 8 Complete fields and valuation rings

18.785 Number theory I Fall 2017 Lecture #8 10/02/2017 8 Complete fields and valuation rings In order to make further progress in our investigation of finite extensions L/K of the fraction field K of a

### Rings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.

Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over

### The Chinese Remainder Theorem

The Chinese Remainder Theorem R. C. Daileda February 19, 2018 1 The Chinese Remainder Theorem We begin with an example. Example 1. Consider the system of simultaneous congruences x 3 (mod 5), x 2 (mod

### 1 Overview and revision

MTH6128 Number Theory Notes 1 Spring 2018 1 Overview and revision In this section we will meet some of the concerns of Number Theory, and have a brief revision of some of the relevant material from Introduction

### Lemma 1.1. The field K embeds as a subfield of Q(ζ D ).

Math 248A. Quadratic characters associated to quadratic fields The aim of this handout is to describe the quadratic Dirichlet character naturally associated to a quadratic field, and to express it in terms

### LECTURE NOTES IN CRYPTOGRAPHY

1 LECTURE NOTES IN CRYPTOGRAPHY Thomas Johansson 2005/2006 c Thomas Johansson 2006 2 Chapter 1 Abstract algebra and Number theory Before we start the treatment of cryptography we need to review some basic

### 4 Number Theory and Cryptography

4 Number Theory and Cryptography 4.1 Divisibility and Modular Arithmetic This section introduces the basics of number theory number theory is the part of mathematics involving integers and their properties.

### Solutions to Problem Set 4 - Fall 2008 Due Tuesday, Oct. 7 at 1:00

Solutions to 8.78 Problem Set 4 - Fall 008 Due Tuesday, Oct. 7 at :00. (a Prove that for any arithmetic functions f, f(d = f ( n d. To show the relation, we only have to show this equality of sets: {d

### Know the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element.

The first exam will be on Monday, June 8, 202. The syllabus will be sections. and.2 in Lax, and the number theory handout found on the class web site, plus the handout on the method of successive squaring

### MATH Fundamental Concepts of Algebra

MATH 4001 Fundamental Concepts of Algebra Instructor: Darci L. Kracht Kent State University April, 015 0 Introduction We will begin our study of mathematics this semester with the familiar notion of even

### Solutions to Assignment 1

Solutions to Assignment 1 Question 1. [Exercises 1.1, # 6] Use the division algorithm to prove that every odd integer is either of the form 4k + 1 or of the form 4k + 3 for some integer k. For each positive

### NOTES ON SIMPLE NUMBER THEORY

NOTES ON SIMPLE NUMBER THEORY DAMIEN PITMAN 1. Definitions & Theorems Definition: We say d divides m iff d is positive integer and m is an integer and there is an integer q such that m = dq. In this case,

### Solutions to Practice Final

s to Practice Final 1. (a) What is φ(0 100 ) where φ is Euler s φ-function? (b) Find an integer x such that 140x 1 (mod 01). Hint: gcd(140, 01) = 7. (a) φ(0 100 ) = φ(4 100 5 100 ) = φ( 00 5 100 ) = (

### Q 2.0.2: If it s 5:30pm now, what time will it be in 4753 hours? Q 2.0.3: Today is Wednesday. What day of the week will it be in one year from today?

2 Mod math Modular arithmetic is the math you do when you talk about time on a clock. For example, if it s 9 o clock right now, then it ll be 1 o clock in 4 hours. Clearly, 9 + 4 1 in general. But on a

### 1 Absolute values and discrete valuations

18.785 Number theory I Lecture #1 Fall 2015 09/10/2015 1 Absolute values and discrete valuations 1.1 Introduction At its core, number theory is the study of the ring Z and its fraction field Q. Many questions

### PRIME NUMBERS YANKI LEKILI

PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers: 1,2,..., These are constructed using Peano axioms. We will not get into the philosophical questions related to this and simply assume

### x = π m (a 0 + a 1 π + a 2 π ) where a i R, a 0 = 0, m Z.

ALGEBRAIC NUMBER THEORY LECTURE 7 NOTES Material covered: Local fields, Hensel s lemma. Remark. The non-archimedean topology: Recall that if K is a field with a valuation, then it also is a metric space

### 5 Group theory. 5.1 Binary operations

5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1

### 2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31

Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15

### ax b mod m. has a solution if and only if d b. In this case, there is one solution, call it x 0, to the equation and there are d solutions x m d

10. Linear congruences In general we are going to be interested in the problem of solving polynomial equations modulo an integer m. Following Gauss, we can work in the ring Z m and find all solutions to

### Elementary Number Theory MARUCO. Summer, 2018

Elementary Number Theory MARUCO Summer, 2018 Problem Set #0 axiom, theorem, proof, Z, N. Axioms Make a list of axioms for the integers. Does your list adequately describe them? Can you make this list as

### MATH 310: Homework 7

1 MATH 310: Homework 7 Due Thursday, 12/1 in class Reading: Davenport III.1, III.2, III.3, III.4, III.5 1. Show that x is a root of unity modulo m if and only if (x, m 1. (Hint: Use Euler s theorem and

### 2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?

Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative

### (1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d

The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers

### Fundamentals of Pure Mathematics - Problem Sheet

Fundamentals of Pure Mathematics - Problem Sheet ( ) = Straightforward but illustrates a basic idea (*) = Harder Note: R, Z denote the real numbers, integers, etc. assumed to be real numbers. In questions

### 4 PRIMITIVE ROOTS Order and Primitive Roots The Index Existence of primitive roots for prime modulus...

PREFACE These notes have been prepared by Dr Mike Canfell (with minor changes and extensions by Dr Gerd Schmalz) for use by the external students in the unit PMTH 338 Number Theory. This booklet covers