Quadratic residues and the Legendre symbol
1 Working in Z7, find all the solutions to x2 = 0, 1, 2, 3, 4, 5, 6 respectively.
2 In Z7, which of the equations
have no solution, have one solution, have two solutions?
3 The perfect squares in M7 are called the quadratic residues modulo 7. Exhibit the mapping of M7 given by x → x2. Does the set of images under this mapping form a subgroup of M7?
4 Use table 3.2 to find the quadratic residues modulo 5, 11, 12 and 13. For which of these moduli do the quadratic residues form a subgroup of Mn ? For which of these values of n is the mapping of Mn given by x → x2, two to one?
5 State the number of quadratic residues modulo 3, 5, 7, 11, 13 and 17 respectively. Predict the number of quadratic residues modulo p (an odd prime).
6 If x2 ≡ y2 (mod p), does it follow that either x = y (mod p) or x =-y (modp), when p is a prime number?
7 For any prime p, determine the elements of Mp which are mapped to 1 under the mapping x → x2. Can you say how many elements of Mp are mapped to any other quadratic residue under this mapping?
8 A number in Mn which is not a square is called a quadratic nonresidue modulo n. Find three quadratic non-residues modulo 12.
9 Find the cube of all the quadratic residues modulo 7, and also the cube of all the quadratic non-residues.
10 What does Fermat's theorem imply about the number of roots of x6=1 (mod 7)?
What can you deduce about the cube of a quadratic residue modulo 7?
By factorising x6- 1 and using Lagrange's theorem, q 3.57, deduce that at most three cubes are congruent to 1 (mod 7) and at most three cubes are congruent to —1 (mod 7).
11 Using the method of q 10, determine how many fifth powers are congruent to 1 (mod 11) and how many fifth powers are congruent to-1 (mod 11). Does the value of the fifth power determine whether the number is a quadratic residue or not?
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.