• Aucun résultat trouvé

give a proof of quadratic reciprocity in the form(pq)(qp

N/A
N/A
Info
Télécharger
Protected

Academic year: 2022

Partager "give a proof of quadratic reciprocity in the form(pq)(qp"

Copied!
1
0
0

Chargement.... (Voir le texte intégral maintenant)

Texte intégral

(1)

QUIZ

MATH-UA 0248-001 THEORY OF NUMBERS November, 9, 2020, 3.30-4.45pm

If doing at a different time, choose 1h15 time slot

CLASS NOTES, BOOKS, NOTES FROM THE RECITATION ARE AUTHORIZED

Submit your quiz by email as the homework

1. give the definition of a quadratic residue mod p, where pis a prime number 2. give the definition of the Legendre symbol

3. formulate Euler’s criterion (give the statement, no proof)

4. formulate and prove Gauss criterion (give the statement and the proof) 5. give a proof of quadratic reciprocity in the form(pq)(qp) = (−1)p−12 q−12 wherep

and q are odd primes.

6. why there is no primitive roots mod 8?

7. give a proof of Lagrange theorem: Let p be a prime number. Let f(x) = anxn+an−1xn−1+. . .+a1x+a0,

p - an be a polynomial of degree n. Then the congruence f(x) ≡ 0 (mod p) has at mostn incongruent solutions mod p.

8. give the idea of the proof of the fact that there are primitive roots mod p (Explain what is the main idea, in your opinion. You are free to choose to provide more or less details).

1

Références

Télécharger maintenant ( PDF - 1 Page - 109.53 KB )

Documents relatifs

We hope they will give a fair idea of the type of research presently being done in the Northeast

Thus, despite the work undertaken by our Indian colleagues over the last forty years, a great deal of research is needed to at least understand the basic anthropological

View of Are we still cartesians? On the idea of life

Isto não acontece porque careçam de órgãos, pois as pegas e os papagaios podem proferir palavras, mas não falar como nós, isto é, testemunhando que pensam o que dizem, ao passo que

We give explicit com- putations in the case of the quadratic branching mechanism

Recently, some authors studied the coalescent process (or genealogical tree) of random size varying population; in this direction see [40] and [32] for branch- ing process, [27]

From the Idea of the Good to some Ideas of Goodman

In spite ofthe ontological connotations of the labels 'platonism' and even 'Platonism', the most important thrust of Plato's thinking, whether he was talking about Forms and

4 The proof of the main theorem

Wang, Error analysis of the semi-discrete local discontinuous Galerkin method for compressible miscible displacement problem in porous media, Applied Mathematics and Computation,

ON THE DEKSITY OF THE ABU-KDAKT NUMBERS 1. The object of this paper is to give a proof that the quotient A(yz),% tends to a limit as n + CCI; where A

It will be seen that the method used in this paper leads immediately to a much better result than o (n/log2 n) for thenumber ofprimitive abundant numbers not

The proof turns out to be a fairly straightforward extension of the proof of the main result in [4]

In this note, we prove that the density of the free additive convolution of two Borel probability measures supported on R whose Cauchy transforms behave well at innity can be

3 Details of the proof of the Liouville theorem

for the heat equation (that is δ = 0 in (2); there, the authors prove a Liouville Theorem which turns to be the trivial case of the Liouville Theorem proved by Merle and Zaag in