An introduction to the p-adic absolute value Ellen Eischen 1 http://eischen.web.unc.edu March 13, 2015 1Partially supported by NSF Grant DMS-1249384. E. Eischen (UNC) p-adic absolute value March 13, 2015 1 / 46 (This was answered by Paul Monsky in an issue of the MAA's American Mathematical Monthly in 1970.) Question Can you dissect a square into an odd number of triangles of equal area? Figure : Equidissection (https://simpletonsymposium.files.wordpress. com/2013/03/monsky-even-squares.jpg) E. Eischen (UNC) p-adic absolute value March 13, 2015 2 / 46 Question Can you dissect a square into an odd number of triangles of equal area? Figure : Equidissection (https://simpletonsymposium.files.wordpress. com/2013/03/monsky-even-squares.jpg) (This was answered by Paul Monsky in an issue of the MAA's American Mathematical Monthly in 1970.) E. Eischen (UNC) p-adic absolute value March 13, 2015 2 / 46 (This was answered by Andrew Wiles, who proved Fermat's Last Theorem in the mid-1990s.) Figure : Andrew Wiles (http://www.simonsingh.net) Question Let n ≥ 3 be an integer. Can you find nonzero integers a, b, and c such that an + bn = cn? E. Eischen (UNC) p-adic absolute value March 13, 2015 3 / 46 Question Let n ≥ 3 be an integer. Can you find nonzero integers a, b, and c such that an + bn = cn? (This was answered by Andrew Wiles, who proved Fermat's Last Theorem in the mid-1990s.) Figure : Andrew Wiles (http://www.simonsingh.net) E. Eischen (UNC) p-adic absolute value March 13, 2015 3 / 46 Answer Their answers are both \no," and the proofs both use the p-adic absolute value. Question What do these two questions have in common? E. Eischen (UNC) p-adic absolute value March 13, 2015 4 / 46 Question What do these two questions have in common? Answer Their answers are both \no," and the proofs both use the p-adic absolute value. E. Eischen (UNC) p-adic absolute value March 13, 2015 4 / 46 Outline 1 Introduction to the p-adic absolute value 2 Significance 3 Dissecting squares E. Eischen (UNC) p-adic absolute value March 13, 2015 5 / 46 The usual absolute value ⎧ ⎪x x ≥ 0 SxS = ⎨ ⎪ ⎩⎪−1 × x else for each real number x. E. Eischen (UNC) p-adic absolute value March 13, 2015 6 / 46 and SaS = 0 if and only if a = 0. 2 Sa × bS = SaS × SbS for all real numbers a and b, i.e. the absolute value is multiplicative 3 Sa + bS ≤ SaS + SbS (triangle inequality) Key properties of absolute value Let a and b be real numbers. Then: 1 SaS ≥ 0 E. Eischen (UNC) p-adic absolute value March 13, 2015 7 / 46 2 Sa × bS = SaS × SbS for all real numbers a and b, i.e. the absolute value is multiplicative 3 Sa + bS ≤ SaS + SbS (triangle inequality) Key properties of absolute value Let a and b be real numbers. Then: 1 SaS ≥ 0 and SaS = 0 if and only if a = 0. E. Eischen (UNC) p-adic absolute value March 13, 2015 7 / 46 3 Sa + bS ≤ SaS + SbS (triangle inequality) Key properties of absolute value Let a and b be real numbers. Then: 1 SaS ≥ 0 and SaS = 0 if and only if a = 0. 2 Sa × bS = SaS × SbS for all real numbers a and b, i.e. the absolute value is multiplicative E. Eischen (UNC) p-adic absolute value March 13, 2015 7 / 46 Key properties of absolute value Let a and b be real numbers. Then: 1 SaS ≥ 0 and SaS = 0 if and only if a = 0. 2 Sa × bS = SaS × SbS for all real numbers a and b, i.e. the absolute value is multiplicative 3 Sa + bS ≤ SaS + SbS (triangle inequality) E. Eischen (UNC) p-adic absolute value March 13, 2015 7 / 46 Definition −k −k The p-adic absolute value of a is p , i.e. SaSp = p . (If a = 0, SaSp ∶= 0.) Example p = 2 1 2 = 21 ⋅ 1 3 3 = 20 ⋅ 1 3 3~14 = 2−1 ⋅ 7 p-adic absolute value Let p be a prime number. Let a be a rational number. Write c a = pk d with k an integer and c and d integers such that p does not divide cd. E. Eischen (UNC) p-adic absolute value March 13, 2015 8 / 46 Example p = 2 1 2 = 21 ⋅ 1 3 3 = 20 ⋅ 1 3 3~14 = 2−1 ⋅ 7 p-adic absolute value Let p be a prime number. Let a be a rational number. Write c a = pk d with k an integer and c and d integers such that p does not divide cd. Definition −k −k The p-adic absolute value of a is p , i.e. SaSp = p . (If a = 0, SaSp ∶= 0.) E. Eischen (UNC) p-adic absolute value March 13, 2015 8 / 46 p-adic absolute value Let p be a prime number. Let a be a rational number. Write c a = pk d with k an integer and c and d integers such that p does not divide cd. Definition −k −k The p-adic absolute value of a is p , i.e. SaSp = p . (If a = 0, SaSp ∶= 0.) Example p = 2 1 2 = 21 ⋅ 1 3 3 = 20 ⋅ 1 3 3~14 = 2−1 ⋅ 7 E. Eischen (UNC) p-adic absolute value March 13, 2015 8 / 46 (Recall: T2k c T = 2−k , if 2 does not divide cd.) d 2 0 S1S2 = S2 ⋅ 1~1S2 = 1 1 S2S = S21 ⋅ 1~1S = 2 2 2 0 S3S2 = S2 ⋅ 3~1S2 = 1 1 S4S = S22 ⋅ 1~1S = 2 2 4 Find the 2-adic absolute value of 3~14. −1 S3~14S2 = S2 ⋅ 3~7S2 = 2 Examples of 2-adic absolute values Find the 2-adic absolute value of each of the first four integers. E. Eischen (UNC) p-adic absolute value March 13, 2015 9 / 46 0 S1S2 = S2 ⋅ 1~1S2 = 1 1 S2S = S21 ⋅ 1~1S = 2 2 2 0 S3S2 = S2 ⋅ 3~1S2 = 1 1 S4S = S22 ⋅ 1~1S = 2 2 4 Find the 2-adic absolute value of 3~14. −1 S3~14S2 = S2 ⋅ 3~7S2 = 2 Examples of 2-adic absolute values Find the 2-adic absolute value of each of the first four integers. (Recall: T2k c T = 2−k , if 2 does not divide cd.) d 2 E. Eischen (UNC) p-adic absolute value March 13, 2015 9 / 46 1 S2S = S21 ⋅ 1~1S = 2 2 2 0 S3S2 = S2 ⋅ 3~1S2 = 1 1 S4S = S22 ⋅ 1~1S = 2 2 4 Find the 2-adic absolute value of 3~14. −1 S3~14S2 = S2 ⋅ 3~7S2 = 2 Examples of 2-adic absolute values Find the 2-adic absolute value of each of the first four integers. (Recall: T2k c T = 2−k , if 2 does not divide cd.) d 2 0 S1S2 = S2 ⋅ 1~1S2 = 1 E. Eischen (UNC) p-adic absolute value March 13, 2015 9 / 46 0 S3S2 = S2 ⋅ 3~1S2 = 1 1 S4S = S22 ⋅ 1~1S = 2 2 4 Find the 2-adic absolute value of 3~14. −1 S3~14S2 = S2 ⋅ 3~7S2 = 2 Examples of 2-adic absolute values Find the 2-adic absolute value of each of the first four integers. (Recall: T2k c T = 2−k , if 2 does not divide cd.) d 2 0 S1S2 = S2 ⋅ 1~1S2 = 1 1 S2S = S21 ⋅ 1~1S = 2 2 2 E. Eischen (UNC) p-adic absolute value March 13, 2015 9 / 46 1 S4S = S22 ⋅ 1~1S = 2 2 4 Find the 2-adic absolute value of 3~14. −1 S3~14S2 = S2 ⋅ 3~7S2 = 2 Examples of 2-adic absolute values Find the 2-adic absolute value of each of the first four integers. (Recall: T2k c T = 2−k , if 2 does not divide cd.) d 2 0 S1S2 = S2 ⋅ 1~1S2 = 1 1 S2S = S21 ⋅ 1~1S = 2 2 2 0 S3S2 = S2 ⋅ 3~1S2 = 1 E. Eischen (UNC) p-adic absolute value March 13, 2015 9 / 46 Find the 2-adic absolute value of 3~14. −1 S3~14S2 = S2 ⋅ 3~7S2 = 2 Examples of 2-adic absolute values Find the 2-adic absolute value of each of the first four integers. (Recall: T2k c T = 2−k , if 2 does not divide cd.) d 2 0 S1S2 = S2 ⋅ 1~1S2 = 1 1 S2S = S21 ⋅ 1~1S = 2 2 2 0 S3S2 = S2 ⋅ 3~1S2 = 1 1 S4S = S22 ⋅ 1~1S = 2 2 4 E. Eischen (UNC) p-adic absolute value March 13, 2015 9 / 46 −1 S3~14S2 = S2 ⋅ 3~7S2 = 2 Examples of 2-adic absolute values Find the 2-adic absolute value of each of the first four integers. (Recall: T2k c T = 2−k , if 2 does not divide cd.) d 2 0 S1S2 = S2 ⋅ 1~1S2 = 1 1 S2S = S21 ⋅ 1~1S = 2 2 2 0 S3S2 = S2 ⋅ 3~1S2 = 1 1 S4S = S22 ⋅ 1~1S = 2 2 4 Find the 2-adic absolute value of 3~14. E. Eischen (UNC) p-adic absolute value March 13, 2015 9 / 46 Examples of 2-adic absolute values Find the 2-adic absolute value of each of the first four integers. (Recall: T2k c T = 2−k , if 2 does not divide cd.) d 2 0 S1S2 = S2 ⋅ 1~1S2 = 1 1 S2S = S21 ⋅ 1~1S = 2 2 2 0 S3S2 = S2 ⋅ 3~1S2 = 1 1 S4S = S22 ⋅ 1~1S = 2 2 4 Find the 2-adic absolute value of 3~14.