Quadratic Integers: Understanding the Theorem and Proving 32 = ab in Q[sqrt -1]

[squelchy451]

For the theorem that states that in quadratic field Q[sqrt d], if d is congruent to 1 mod 4, then it is in the form (a + b sqrt d)/2 and if it's not, it's in the form a + b sqrt d where a and b are rational integers, is it saying that if a and b are rational integers and the quadratic number are in the form according to its congruency mod 4, then the quadratic number is an integer?

Also, how would you prove that if 32 = ab where a and b are relatively prime quadratic integers in Q[sqrt -1], a = e(g^2) where e is a unit and g is a quadratic integer in Q [sqrt -1].

 

[ramsey2879]

For the theorem that states that in quadratic field Q[sqrt d], if d is congruent to 1 mod 4, then it is in the form (a + b sqrt d)/2 and if it's not, it's in the form a + b sqrt d where a and b are rational integers, is it saying that if a and b are rational integers and the quadratic number are in the form according to its congruency mod 4, then the quadratic number is an integer?

Also, how would you prove that if 32 = ab where a and b are relatively prime quadratic integers in Q[sqrt -1], a = e(g^2) where e is a unit and g is a quadratic integer in Q [sqrt -1].

When you have the product of two quadratic integers equal an integer, look first for two of the form A +/- B(sqrt -1). That is one use negative B and the other uses positive B so that the quadratic part cancels out.

 

[squelchy451]

On a side note, are a + b sqrt d and a - b sqrt d where a, b, and d are rational integers (and d is not a perfect square) relatively prime?

 

[ramsey2879]

On a side note, are a + b sqrt d and a - b sqrt d where a, b, and d are rational integers (and d is not a perfect square) relatively prime?

Well if 2, b and sqrt d each are relatively prime to a, then I would say that the two factors are relatively prime but I am not sure. However, I think you have to look to cancel the quadratic part in another way since 4 + 4i and 4 - 4i are not relatively prime. Maybe try playing around with numbers 1,16 and i in lieu of 4 and i.

 

[blue_raver22]

The theorem that you mentioned is known as the "Gaussian Integers Theorem" and it states that in a quadratic field Q[sqrt d], where d is a positive integer, if d is congruent to 1 mod 4, then any element can be written in the form (a + b sqrt d)/2, where a and b are rational integers. If d is not congruent to 1 mod 4, then any element can be written in the form a + b sqrt d, where a and b are rational integers.

This theorem does not necessarily state that if a and b are rational integers and the quadratic number is in the form according to its congruency mod 4, then the quadratic number is an integer. It simply provides a way to express elements in a quadratic field in a specific form, which can be useful in certain calculations and proofs.

To prove that 32 = ab where a and b are relatively prime quadratic integers in Q[sqrt -1], we can use the unique factorization property of Gaussian integers. This property states that any Gaussian integer can be expressed as a product of a unit and prime Gaussian integers. In this case, we have 32 = ab, where a and b are relatively prime, which means that they do not share any common factors (other than units).

We can then use this property to write a and b as the product of units and prime Gaussian integers. Let a = e1p1p2...pn and b = e2q1q2...qm, where e1 and e2 are units, p1, p2, ..., pn are prime Gaussian integers and q1, q2, ..., qm are also prime Gaussian integers.

Since 32 = ab, we can equate the two expressions and get e1e2p1p2...pnq1q2...qm = 32. Since 32 is a rational integer, e1e2 must also be a rational integer. This means that e1 and e2 are actually units in the field Q[sqrt -1]. Therefore, we can write a = e(g^2) and b = g, where e is a unit and g is a quadratic integer in Q[sqrt -1].

In conclusion, the Gaussian Integers Theorem provides a useful way to express elements in a quadratic field and the unique factorization property can be used to prove statements about Gaussian integers.