Roots of Unity

6 Key excerpts on "Roots of Unity"

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  Abstract Algebra

  • Paul B. Garrett(Author)
  • 2007(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  19. Roots of Unity 19.1 Another proof of cyclicness 19.2 Roots of Unity 19.3 with Roots of Unity adjoined 19.4 Solution in radicals, Lagrange resolvents 19.5 Quadratic fields, quadratic reciprocity 19.6 Worked examples 19.1 Another proof of cyclicness Earlier, we gave a more complicated but more elementary proof of the following theorem, using cyclotomic polynomials. There is a cleaner proof using the structure theorem for finite abelian groups, which we give now. Thus, this result is yet another corollary of the structure theory for finitely-generated free modules over PIDs. 19.1.1 Theorem: Let G be a finite subgroup of the multiplicative group k × of a field k . Then G is cyclic. Proof: By the structure theorem, applied to abelian groups as -modules, G ≈ /d 1 ⊕ . . . ⊕ /d n where the integers d i have the property 1 < d 1 | . . . | d n and no elementary divisor d i is 0 (since G is finite). All elements of G satisfy the equation x d t = 1 The argument using cyclotomic polynomials is wholesome and educational, too, but is much grittier than the present argument. 243 244 Roots of Unity By unique factorization in k [ x ], this equation has at most d t roots in k . Thus, there can be only one direct summand, and G is cyclic. /// 19.1.2 Remark: Although we will not need to invoke this theorem for our discussion just below of solutions of equations x n = 1 one might take the viewpoint that the traditional pictures of these solutions as points on the unit circle in the complex plane are not at all misleading about more general situations. 19.2 Roots of Unity An element ω in any field k with the property that ω n = 1 for some integer n is a root of unity . For positive integer n , if ω n = 1 and ω t = 1 for positive integers t < n , then ω is a primitive n th root of unity. Note that μ n = { α ∈ k × : α n = 1 } is finite since there are at most n solutions to the degree n equation x n = 1 in any field. This group is known to be cyclic , by at least two proofs.

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  Advanced Number Theory with Applications

  • Richard A. Mollin(Author)
  • 2009(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  . . } ( the natural numbers ), ζ n denotes a primitive n th root of unity, which is a root of x n -1, but not a root of x d -1 for any natural number d < n . Example 1.2 ζ 3 = ( -1 + √ -3) / 2 is a primitive cube root of unity since it is a root of x 3 -1, but clearly not a root of x 2 -1 or x -1. A special kind of algebraic integer is given in the following. Example 1.3 Numbers of the form z 0 + z 1 ζ n + z 2 ζ 2 n + · · · + z n -1 ζ n -1 n , for z j ∈ Z , are called cyclotomic integers of order n . Definition 1.2, in turn, is a special case of the following. Definition 1.3 Units An element α in a commutative ring R with identity 1 R is called a unit in R when there is a β ∈ R such that αβ = 1 R . The multiplicative group of units in R is denoted by U R . Example 1.4 In Z [ √ 2] = R , 1 + √ 2 is a unit since (1 + √ 2)( -1 + √ 2) = 1 R = 1 . Definition 1.4 Algebraic Numbers and Number Fields An algebraic number, α , of degree d ∈ N is a root of a monic polynomial in Q [ x ] of degree d and not the root of any polynomial in Q [ x ] of degree less than d . In other words, an algebraic number is the root of an irreducible polynomial of degree d over Q . An algebraic number field , or simply number field , is of the form F = Q ( α 1 , α 2 , . . . , α n ) ⊆ C for n ∈ N where α j for j = 1 , 2 , . . . , n are algebraic numbers. Denote the subfield of C consisting of all algebraic numbers by Q , and the set of all algebraic integers in Q by A . An algebraic number of degree d ∈ N over a number field F is the root of an irreducible polynomial of degree d over F . 1.1. Algebraic Number Fields 3 Remark 1.1 If F is a simple extension , namely of the form Q ( α ), for an algebraic number α , then we may consider this as a vector space over Q , in which case we may say that Q ( α ) has dimension d over Q having basis { 1 , α, .

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  Elementary Algebra

  • Mark D. Turner, Charles P. McKeague(Authors)
  • 2017(Publication Date)
  • XYZ Textbooks

    (Publisher)

  1 1 1 1 1 1 1 1 1 1 1 1 1 2 √ 3 √ 4 √ 5 √ 6 √ 7 √ 8 √ 9 √ 10 √ 11 √ 12 √ 13 √ Chapter Outline 8.1 Definitions and Common Roots 8.2 Simplified Form and Properties of Radicals 8.3 Addition and Subtraction of Radical Expressions 8.4 Multiplication of Radicals 8.5 Division of Radicals 8.6 Equations Involving Radicals 8 521 iStockphoto.com © Katarzyna Krawlec T he diagram above is called the spiral of roots . The spiral of roots mimics the shell of the chambered nautilus, an animal that has survived largely un-changed for millions of years. The mathematical diagram is constructed using the Pythagorean theorem, which we introduced in Chapter 6. The spiral of roots gives us a way to visualize positive square roots, one of the topics we will cover in this chapter. Table 1 gives the lengths of the diagonals in the spiral of roots, accurate to the nearest hundredth. If we take each of the diagonals in the spiral of roots and place it above the corresponding whole number on the x -axis and then connect the tops of all these segments with a smooth curve, we have the graph shown in Figure 1. This curve is also the graph of the equation y = √ — x . FIGURE 1 1 2 3 4 5 6 7 8 9 10 0 4 1 2 3 1 3 2 6 5 4 9 8 7 10 TABLE 1 Approximate Length of Diagonals Positive Square Number Root 1 1 2 1.41 3 1.73 4 2 5 2.24 6 2.45 7 2.65 8 2.83 9 3 10 3.16 Roots and Radical Expressions 522 Success Skills Never mistake activity for achievement. — John Wooden, legendary UCLA basketball coach You may think that the John Wooden quote above has to do with being productive and efficient, or using your time wisely, but it is really about being honest with yourself. I have had students come to me after failing a test saying, I can’t understand why I got such a low grade after I put so much time in studying. One student even had help from a tutor and felt she understood everything that we covered.

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  Fearless Symmetry

  Exposing the Hidden Patterns of Numbers - New Edition

  • Avner Ash, Robert Gross(Authors)
  • 2008(Publication Date)
  • Princeton University Press

    (Publisher)

  Sometimes “root” means a square root or cube root, and so on, and sometimes it means a root of a more general polynomial. You should be able to tell which root is meant from the context. It is helpful to symbolize the polynomial we are studying by a single letter, say p. If we want to remember the name of the variable, we can write p(x). For instance, p(x) might denote the polynomial x 3 + x − 2. Then equation (6.2) can be written p(x) = 0. This looks like functional notation, and it is. If x is a variable, p(x) just stands for the polynomial p, but if a is a number, p(a) stands for the number you get by substituting a for x in p. For instance, if p(x) = x 3 + x − 2 then p(0) = 0 3 + 0 − 2 = −2, p(1) = 1 3 + 1 − 2 = 0, and in general, p(a) = a 3 + a − 2. For example, consider the variety of solutions, call it S, to the equation p(x) = 0, where p(x) is the polynomial discussed in the pre- ceding paragraph. That is, if A is any number system, describe the set S(A), which is the set of all elements a of A such that p(a) = 0. We can use simple algebra and some guessing to find S(Z): We have just seen that p(1) = 0. So S(A) contains the number 1. High- school algebra now tells us that we can divide p(x) by x − 1 and we are guaranteed that it will go in without remainder. Doing that we get the quotient x 2 + x + 2. 62 CHAPTER 6 In other words, p(x) = x 3 + x − 2 = (x − 1)(x 2 + x + 2). Now for any integer a, p(a) = 0 if and only if (a − 1)(a 2 + a + 2) = 0, because p(a) = (a − 1)(a 2 + a + 2). A product of two integers is 0 only if one or both of them is 0. (This is true in any number system that we will use in this book.) So if we try to solve (a − 1)(a 2 + a + 2) = 0, we see that either a = 1 (which we already knew was a possibility) or else a 2 + a + 2 = 0. The quadratic formula 6 will tell us what all the solutions of the quadratic equation are: a = −1± √ 1−8 2 . But the negative number −7 has no real square root, let alone an integer square root.

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  Numbers and Proofs

  • Reg Allenby(Author)
  • 1997(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  NUMBERS AND PROOFS sin {§ The above is only the tip of the iceberg. We can go on to consider general functions of a complex variable, their derivatives and their integrals over contours in the complex plane. Such activity has applications in many areas of applied mathematics - even in civil engineering (where it is important in studying fluid seepage under dams). Back in pure mathematics, an estimate for the number of primes less than the integer n can be obtained via the Prime Number Theorem which may be proved by considering complex integrals and logarithms. But these are other stories. Summary Following their introduction by means of a 'problem', complex numbers were used informally by many mathematicians before it was deemed necessary to try to put them on a firmer footing. Simply defining a complex number to be 'something of the form a + ib (or a + M) where a and b are real numbers and i represents y/— Γ can lead to (temporary) difficulties - which can be eliminated by using Hamilton's definition of complex numbers using ordered pairs. Sticking to the a + ib notation we illustrated the 'trick' for simplifying a fraction of two complex numbers, and observed that the complex numbers satisfy the basic rules of arithmetic. We then stated the very important Fundamental Theorem of Algebra and deduced that, in each polynomial equation with real coefficients, the roots appear in complex conjugate pairs. The important concepts of modulus and argument were introduced via polar representation of complex numbers, and their more prominent properties - in particular the triangle inequality - were established. The neat polar form of complex multiplication leads, naturally, to De Moivre's theorem with its applications to finding roots and obtaining trigonometrical identities.

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  Sixth Form Pure Mathematics

  Volume 2

  • C. Plumpton, W. A. Tomkys(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  Each of the complex cube Roots of Unity is thus the square of the other. The three roots are usually denoted by 1, ω, ω 2 . (i) Because 1, ω, ω 2 are the roots of x 3 — 1 = 0, a cubic equation in x in which the coefficient of x 2 is zero, .·. 1 -f ω 4- ω 2 = 0. (This result also follows directly from the values of ω, ω 2 given above.) 198 PURE MATHEMATICS (ii) ω 3 « + 1 = ω , ω 3 η + 2 = ω 2 for n = 0, 1, 2, . . . . (iii) The cube roots of any real number N (say) are N llz , Ν 1ΐ3 ω and Ν 1ΐ3 ω 2 where N llz denotes the real cube root of N. Examples, (i) Show that a 3 + b z + c 3 - 3 a b c = (a + b + c) (a + ω 6 + ω 2 c) (a -f ω 2 6 + ω c). (a + ω 6 -f ω 2 c) (a + ω 2 6 + ω c) = a 2 + ω 2 a b + ω a c + ω « δ + ω 3 6 2 4-ω 2 b c + ω 2 a c + ω 4 b c + ω 3 c 2 EEa 2 + & 2 + c 2 + a6(co 2 + ω ) + 6c( odd. 2 (i) lin = 3m, ( -ω ) Λ = -(co 3 ) m = -1 and £ n = _ — = _ 2 ω . 1 ~r (ί> 1 — ω (ii) If n = 3m + 1, ( -ω ) η = ω 3 ™. ω = ω and Ä n = — ---, Λ 1 + ω 2 1 (in) Ifn = 3m + 2,(-cü) w =-co 3 m . ω 2 = -ω 2 and £„ = -- = -. When m is even. (i) If w = 3 m, (-ω) η = (ω 8 Γ = 1 and S H = 0. (ii) I f n = 3 m H -l , ( -oe ) n = -ω 3 ™. ω = -ω and S n = y ^ - = 1. (iii) If Λ = 3m -f 2, ( -oj) n = co 3w . ω 2 = OJ 2 1 2 and S n = -- — = 1 — ω . n 1 + ω

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