Daniel Madden

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Scholarly Contributions

Books

• Madden, D., & Aubrey, J. (2018). Introduction to Proof through Real Analysis. Wiley Publications.
• Madden, D., & Aubrey, J. (2017). Introduction to Proof through Real Analysis. Wiley Publications.

Chapters

• Madden, D. (2011). The Case of the ArizonaTeachers Institute. In Perspectives on Deepening Teachers’ Mathematics Content Knowledge:.

Journals/Publications

• Jacobi, L. W., & Madden, D. J. (2008). On a⁴ + b⁴ + c⁴ + d⁴ = (a + b + c + d)⁴. American Mathematical Monthly, 115(3), 220-236.
• Madden, D., & Jacobi, L. (2008). . On a^4+b^4+c^4+d^4=(a+b+c+d)^4. American Mathematical Monthly, Vol.115, p220-236.
• Madden, D. J. (2001). Constructing families of long continued fractions. Pacific Journal of Mathematics, 198(1), 123-147.
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  Abstract: This paper describes a method of constructing an unlimited number of infinite families of continued fraction expansions of the square root of D, an integer. The periods of these continued fractions all have identifiable sub patterns repeated a number of times according to certain parameters. For example, it is possible to construct an explicit family for the square root of D(k, l) where the period of the continued fraction has length 2kl - 2. The method is recursive and additional parameters controlling the length can be added.
• Gomez-Calderon, J., & Madden, D. J. (1988). Polynomials with small value set over finite fields. Journal of Number Theory, 28(2), 167-188.
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  Abstract: Let Kq denote the finite field with q elements and characteristic p. Let f(x) be a monic polynomial of degree d with coeficients in Kq. Let C(f) denote the number of distinct values of f(x) as x ranges over Kq. We easily see that C(f)≥ q-1 d+1 where [{norm of matrix}] is the greatest integer function. A polynomial for which equality in (*) occurs is called a minimum value set polynomial. There is a complete characterization of minimum value set polynomials over arbitrary finite fields with d < √q and C(f) ≥ 3 [see L. Carlitz, D. J. Lewis, W. H. Mills, and E. G. Strauss, Mathematika 8 (1961), 121-130; W. H. Mills, Pacific J. Math. 14 (1964), 225-241]. In this paper we give a complete list of polynomials of degree d4 < q which have a value set of size less than 2q d, twice the minimum possible. If d > 15 then f(x) is one of the following polynomial forms: 1. (a) f(x) = (x + a)d + b, where d | (q - 1), 2. (b) f(x) = ((x + a) d 2 + b)2 + c, where d | (q2 - 1), 3. (c) f(x) = ((x + a)2 + b) d 2 + c, where d | (q2 - 1), or 4. (d) f(x) = Dd,a(x + b) + c, where Dd,a(x) is the Dickson polynomial of degree d, d | (q2 - 1) and a is a 2kth power in Kq2 where d = 2kr, r is odd. The result is obtained by noticing the connection between the size of the value set of a polynomial f(x) and the factorization of the associated substitution polynomial f*(x, y) = f(x) - f(y) in the ring Kq[x, y]. Essentually, we show that C(f) < 2q d implies that f*(x, y) has at least d 2 factors in Kq[x, y], and we determine all the polynomials with such characteristic. © 1988.
• Madden, D. J. (1981). Polynomials and primitive roots in finite fields. Journal of Number Theory, 13(4), 499-514.
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  Abstract: The primitive elements of a finite field are those elements of the field that generate the multiplicative group of k. If f(x) is a polynomial over k of small degree compared to the size of k, then f(x) represents at least one primitive element of k. Also f(x) represents an lth power at a primitive element of k, if l is also small. As a consequence of this, the following results holds. Theorem. Let g(x) be a square-free polynomial with integer coefficients. For all but finitely many prime numbers p, there is an integer a such that g(a) is equivalent to a primitive element modulo p. Theorem. Let l be a fixed prime number and f(x) be a square-free polynomial with integer coefficients with a non-zero constant term. For all but finitely many primes p, there exist integers a and b such that a is a primitive element and f(a) ≡ b1 modulo p. © 1981.
• Madden, D. J., & Vélez, W. Y. (1979). A note on the normality of unramified, abelian extensions of quadratic extensions. Manuscripta Mathematica, 30(4), 343-349.
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  Abstract: Let F, K and L be algebraic number fields such that {Mathematical expression}, [K:F]=2 and [L:K]=n. It is a simple consequence of the class field theory that, if L is an abelian, unramified extension of K and (n,h)=1, where h is the class number of F, then L is normal over F. The purpose of this note is to point out the necessity of the condition (n,h)=1 by constructing for any field F with even class number a tower of fields {Mathematical expression} with [K:F]=2, [L:K]=2 where L is unramified over K, but L is not normal over F. © 1980 Springer-Verlag.
• Madden, D. J. (1978). Arithmetic in generalized Artin-Schreier extensions of k(x). Journal of Number Theory, 10(3), 303-323.
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  Abstract: If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions Kn over k(x) such that [K : k(x)] = pn using the concept of Witt vectors. This is accomplished in the following way; if [β1, β2,..., βn] is a Witt vector over k(x) = K0, then the Witt equation yp • y = β generates a tower of extensions through Ki = Ki-1(yi) where y = [y1, y2,..., yn]. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in Kn. This alternate generation has the form Ki = Ki-1(yi); yip - yi = Bi, where, as a divisor in Ki-1, Bi has the form (Bi) = q Πpjλj. In this form q is prime to Πpjλj and each λj is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field Kn over an algebraically closed field of constants. © 1978.
• Madden, D. J. (1977). Quadratic function fields with invariant class group. Journal of Number Theory, 9(2), 218-228.
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  Abstract: Emil Artin studied quadratic extensions of k(x) where k is a prime field of odd characteristic. He showed that there are only finitely many such extensions in which the ideal class group has exponent two and the infinite prime does not decompose. The main result of this paper is: If K is a quadratic imaginary extension of k(x) of genus G, where k is a finite field of order q, in which the infinite prime of k(x) ramifies, and if the ideal class group has exponent 2, then q = 9, 7, 5, 4, 3, or 2 and G ≤ 1, 1, 2, 2, 4, and 8, respectively. The method of Artin's proof gives G ≤ 13, 9, and 9724 for q = 7, 5, and 3, respectively. If the infinite prime is inert in K, both the methods of this paper and Artin's methods give bounds on the genus that are roughly double those in the ramified case. © 1977.
• Madan, M. L., & Madden, D. J. (1976). Null class groups of algebraic function fields. Mathematische Zeitschrift, 152(1), 59-66.