prime, and G = (Z/pZ)*Let p be aShow that p(a)(b)(c)1 is its own inverse in G.Show that 1 and p - 1 are the only elements of G that are their own inverses(Wilson's Theorem.) Show that (p - 1)! = (p- 1)mod p = -1 mod p

To prove the required properties in the multiplicative group G of integers modulo p, p being a prime…

Answered: prime, and G = (Z/pZ)* Let p be a Show…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 33E: 33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all...

See similar textbooks

Similar questions

30. Prove statement of Theorem : for all integers .
Let p be prime and G the multiplicative group of units Up=[a]Zp[a][0]. Use Lagranges Theorem in G to prove Fermats Little Theorem in the form [a]p=[a] for any aZ. (compare with Exercise 54 in section 2.5) Let p be a prime integer. Prove Fermats Little Theorem: For any positive integer a,apa(modp). (Hint: Use induction on a, with p held fixed.)
49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove that if is a prime integer, then is a factor of for . (From Exercise of Section, it is known that is an integer.) b. Use induction on to prove that if is a prime integer, then is a factor of .
a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)
Use mathematical induction to prove that if a1,a2,...,an are elements of a group G, then (a1a2...an)1=an1an11...a21a11. (This is the general form of the reverse order law for inverses.)
Use generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection with this result, see the discussion of counterexamples in the Appendix.) 1+2n2n for all integers n3
11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .
True or False Label each of the following statements as either true or false. An r-cycle is an even permutation if r is even and an odd permutation if r is odd.
33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .)

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

College Algebra

Algebra

ISBN:

9781305115545

Author:

James Stewart, Lothar Redlin, Saleem Watson

Publisher:

Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

College Algebra

Algebra

ISBN:

9781305115545

Author:

James Stewart, Lothar Redlin, Saleem Watson

Publisher:

Cengage Learning

College Algebra

Algebra

ISBN:

9781337282291

Author:

Ron Larson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

prime, and G = (Z/pZ)* Let p be a Show that p (a) (b) (c) 1 is its own inverse in G. Show that 1 and p - 1 are the only elements of G that are their own inverses (Wilson's Theorem.) Show that (p - 1)! = (p - 1)mod p = -1 mod p