Trying to understand Gauss's Lemma

  • Thread starter Oxymoron
  • Start date
In summary: I still don't understand how floor(p/4) relates to floor((4n+1)/4). Can you help clarify?In summary, the Gauss Lemma states that (2/p) = (-1)^n for any odd prime p. The p odd prime case is when p=2. If p=4n+1 then floor(p/4) is 4 and floor((4n+1)/4)=floor(n+1/4)=5. If p=3(mod 4) then floor((4n+2)/4)=floor(n+2/4)=7 and floor(p/4) is 3.
  • #36
1. Let p=1(mod 8). => p=8n+1

exponent = 8n/4 - floor((8n+1)/4) = 4n - 2n = 2n (EVEN)

2. Let p=-1(mod 8) => p=8n-1

exponent = (8n-2)/2 - floor((8n-1)/4) = 4n - 1 - 2n = 2n-1 (ODD)

3. Let p=3(mod 8) => p=8n+3

exponent = (8n+2)/2 - floor((8n+3)/4) = 4n - 1 - 2n = 2n-1 (ODD)

4. Let p=-3(mod 8) => p=8n-3

exponent = (8n-4)/2 - floor((8n-3)/4) = 4n - 2 - 2n = 2n-2 (EVEN)
 
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  • #37
But shouldn't the first two exponents be EVEN and the last two ODD?
 
  • #38
You have floor((8n-1)/4)=2n, this isn't correct.
 
  • #39
Sorry, where do I have that?

floor((8n-1)/4) = floor(2n-1/4) which is always going to be 1 less than an even number => ODD

n=1 RHS = 1
n=2 RHS = 3
n=3 RHS = 5
etc...
 
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  • #40
Oxymoron said:
2. Let p=-1(mod 8) => p=8n-1

exponent = (8n-2)/2 - floor((8n-1)/4) = 4n - 1 - 2n = 2n-1 (ODD)

(8n-2)/2=4n-1 so

(8n-2)/2 - floor((8n-1)/4) = 4n - 1 - 2n

tells me you have

floor((8n-1)/4) = 2n
 
  • #41
Ah, I see. Well floor((8n-1)/4) is certainly not 2n. ;)

floor((8n-1)/4) = 2n-1 So we should have

4n - 1 - (2n - 1) = 2n => EVEN


And fixing the mistake in the last case yields ODD
 
  • #42
Well, if this is all correct then I have shown what I needed to show. :)
 
<h2>1. What is Gauss's Lemma?</h2><p>Gauss's Lemma is a mathematical theorem named after the famous mathematician Carl Friedrich Gauss. It is used in number theory to determine whether a polynomial with integer coefficients is irreducible over the rational numbers.</p><h2>2. How does Gauss's Lemma work?</h2><p>Gauss's Lemma states that if a polynomial with integer coefficients can be factored into two polynomials with rational coefficients, then it can also be factored into two polynomials with integer coefficients. This allows us to determine whether a polynomial is irreducible over the rational numbers by checking if it can be factored into polynomials with integer coefficients.</p><h2>3. Why is Gauss's Lemma important?</h2><p>Gauss's Lemma is important because it provides a necessary and sufficient condition for determining whether a polynomial is irreducible over the rational numbers. It is also used in the proof of the fundamental theorem of algebra, which states that every polynomial with complex coefficients can be factored into linear factors.</p><h2>4. Can Gauss's Lemma be applied to polynomials with coefficients other than integers?</h2><p>Yes, Gauss's Lemma can be generalized to polynomials with coefficients in any unique factorization domain. This includes polynomials with coefficients in other number fields, such as the Gaussian integers.</p><h2>5. How is Gauss's Lemma related to the Eisenstein criterion?</h2><p>The Eisenstein criterion is a special case of Gauss's Lemma, where the polynomial in question has a prime number as its leading coefficient and all other coefficients are divisible by that prime. In this case, the polynomial is irreducible over the rational numbers. Therefore, the Eisenstein criterion can be seen as an application of Gauss's Lemma.</p>

1. What is Gauss's Lemma?

Gauss's Lemma is a mathematical theorem named after the famous mathematician Carl Friedrich Gauss. It is used in number theory to determine whether a polynomial with integer coefficients is irreducible over the rational numbers.

2. How does Gauss's Lemma work?

Gauss's Lemma states that if a polynomial with integer coefficients can be factored into two polynomials with rational coefficients, then it can also be factored into two polynomials with integer coefficients. This allows us to determine whether a polynomial is irreducible over the rational numbers by checking if it can be factored into polynomials with integer coefficients.

3. Why is Gauss's Lemma important?

Gauss's Lemma is important because it provides a necessary and sufficient condition for determining whether a polynomial is irreducible over the rational numbers. It is also used in the proof of the fundamental theorem of algebra, which states that every polynomial with complex coefficients can be factored into linear factors.

4. Can Gauss's Lemma be applied to polynomials with coefficients other than integers?

Yes, Gauss's Lemma can be generalized to polynomials with coefficients in any unique factorization domain. This includes polynomials with coefficients in other number fields, such as the Gaussian integers.

5. How is Gauss's Lemma related to the Eisenstein criterion?

The Eisenstein criterion is a special case of Gauss's Lemma, where the polynomial in question has a prime number as its leading coefficient and all other coefficients are divisible by that prime. In this case, the polynomial is irreducible over the rational numbers. Therefore, the Eisenstein criterion can be seen as an application of Gauss's Lemma.

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