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Euclidean Domains - Dummit and Foote - Chapter 8 - Section 8.1 - Example on Quadratic

I am reading Dummit and Foote Chapter 8, Section 8.1 - Euclidean DOmains

I am working through Example 2 on page 273 (see attachment)

Example 2 demonstrates that the quadratic integer ring $\displaystyle \mathbb{Z} [ \surd -5 ]$ is not a Euclidean domain.

I can follow the argument down to the point where D&F state (see attachment)

"Multiplying both sdes by $\displaystyle 2 - \surd -5 $ would then imply that $\displaystyle 2 - \surd -5 $ is a multiple of 3 in R, a contradiction"

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I cannot show this point - the mechanics of this fail me... can someone please help

Peter

Re: Euclidean Domains - Dummit and Foote - Chapter 8 - Section 8.1 - Example on Quadr

$\displaystyle 1 \in I = (3, 2 + \sqrt{-5})$ implies $\displaystyle \exists \gamma, \delta \in \mathbb{Z}[\sqrt{-5}]$ such that $\displaystyle 3 \gamma + (2 + \sqrt{-5}) \delta = 1$.

Multiply both sides by $\displaystyle 2 - \sqrt{-5}$, getting:

$\displaystyle 3(2 - \sqrt{-5})\gamma + (2 + \sqrt{-5}) (2 - \sqrt{-5}) \delta = 2 - \sqrt{-5}$, so

$\displaystyle 3(2 - \sqrt{-5})\gamma + 9 \delta = 2 - \sqrt{-5}$, so

$\displaystyle 3 \{(2 - \sqrt{-5})\gamma + 3 \delta \} = 2 - \sqrt{-5}$.

Thus $\displaystyle 3 \alpha = 2 - \sqrt{-5}$, where $\displaystyle \alpha = (2 - \sqrt{-5})\gamma + 3 \delta \in \mathbb{Z}[\sqrt{-5}]$.

Thus $\displaystyle 3$ divides $\displaystyle (2 - \sqrt{-5})$ in $\displaystyle \mathbb{Z}[\sqrt{-5}]$.

But that's impossible because $\displaystyle 3x = 2$ has no solution in $\displaystyle \mathbb{Z}$.

(In detail, if $\displaystyle \alpha = x + y\sqrt{-5}, x, y\in \mathbb{Z}$, then $\displaystyle 3\alpha = 3x + 3y\sqrt{-5}$, so

$\displaystyle 3 \alpha = 2 - \sqrt{-5}$ implies $\displaystyle 3x + 3y\sqrt{-5} = 2 - \sqrt{-5}$ implies $\displaystyle 3x = 2, 3y =-1$.)

Therefore $\displaystyle 1 \notin I$.

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There's a tiny mistake in the proof of Proposition1. It should read "by the Well Ordering of $\displaystyle \mathbb{N}$".

Re: Euclidean Domains - Dummit and Foote - Chapter 8 - Section 8.1 - Example on Quadr

Euclidean domains are a very restrictive class of rings. They are "almost" fields. In particular, they are: unique factorization domains, greatest common divisor domains, and principal ideal domains.

So if a given ring lacks one of these properties, we can conclude it is NOT a Euclidean domain. In this case, D&F choose to show that $\displaystyle \mathbb{Z}[\sqrt{-5}]$ is not a PID.

One can also show R is not a UFD:

9 = 3*3

9 = (2+√(-5))(2-√(-5))

are two distinct factorizations of 9 (that is 3 is not a factor of either 2+√(-5) or 2-√(-5)), which is equivalent to showing that 3 is not prime in R (3 divides a product ab, but divides neither a nor b). However, 3 IS irreducible in R, and in a Euclidean domain "irreducibles = primes" (the same norm N can be used to show that 3 is irreducible:

if 3 = (a+b√(-5))(c+d√(-5)), then N(3) = 9, so we have either:

N(a+b√(-5)) = 1,3 or 9. if N(a+b√(-5)) = 1, then a = ±1, b = 0, in which case a+b√(-5) = ±1 is a unit. A similar proof show c+d√(-5) = ±1 if N(a+√(-5)) = 9. so if both a+b√(-5) and c+d√(-5) are to be non-units, we must have N(a+b√(-5)) = 3. this means a^{2}+5b^{2} = 3, for INTEGERS a,b, so |b| < 1, and is thus 0, and a^{2} = 3 has no integer solution).

In general, it is more convenient to characterize rings by the properties of ideals, rather than elements (by analogy to groups, where we characterize groups by the behavior of normal subgroups: that is, which factor groups we can form from them). In fact, the word "ideal" comes from the term "ideal numbers" which were first studied in quadratic extension rings of the integers (perhaps motivated by a desire to solve Fermat's Last Theorem) as "generalizations" of "prime numbers" in ordinary arithmetic (integers). The general construction is this:

One starts with Q, the rational numbers, and adjoins a root of a quadratic polynomial with integer coefficients, so one gets Q(a). then one considers the sub-ring Z[a]. The ring-theoretic properties depend on a, for some choices we get a Euclidean domain, for some we do not. The general idea is to extend "number theory" to such rings as much as possible. The "euclidean" definition of primes: p is a prime iff p|ab implies p|a or p|b generalizes to a prime ideal: ab in P implies a in P or b in P. If R is a Euclidean domain (the nicest situation), then R is a PID, and the prime ideals P are generated by prime elements p: P = (p).

In the case at hand, the polynomial is x^{2} + 5 in Q[x]. Z[√(-5)] are the "integers" of the field Q(a), where a is a root of that polynomial. because this ring is non-euclidean, "factoring" isn't as helpful as it could be (we cannot say that just because something is irreducible, it is prime, so divisibility arguments can go astray).

Re: Euclidean Domains - Dummit and Foote - Chapter 8 - Section 8.1 - Example on Quadr

Thank you for these posts

Most helpful for those like me engaged in self-study of mathematics

Peter

Re: Euclidean Domains - Dummit and Foote - Chapter 8 - Section 8.1 - Example on Quadr

Deveno ,

thanks for the considerable help

Working through the detail of your post now - your post is much appreciated

Peter

Re: Euclidean Domains - Dummit and Foote - Chapter 8 - Section 8.1 - Example on Quadr

Deveno,

I followed your post except for the following point - you write:

"N(a+b√(-5)) = 1,3 or 9. if N(a+b√(-5)) = 1, then a = ±1, b = 0, in which case a+b√(-5) = ±1 is a unit. A similar proof show c+d√(-5) = ±1 if N(a+√(-5)) = 9.** so if both a+b√(-5) and c+d√(-5) are to be non-units,** we must have N(a+b√(-5)) = 3. this means a2+5b2 = 3, for INTEGERS a,b, so |b| < 1, and is thus 0, and a2 = 3 has no integer solution)."

My question is "Why do both a+b√(-5) and c+d√(-5) have to be non-units?"

[Apologies ... I suspect my question is rather basic ... but I have only just now skimmed the material on UFDs and have not covered them properly]

Peter

Re: Euclidean Domains - Dummit and Foote - Chapter 8 - Section 8.1 - Example on Quadr

in a unique factorization domain, the factorizations are only unique up to units.

for example, in Z, we have: 6 = 2*3 = 1*1*2*3, but we don't really consider these "different" because 1 is a unit. this is the logic behind the rule: "1 is not a prime number".

again, in say, Q[x], when we factor 4x^{2} - 4, we have (2x + 2)(2x - 2), AND (4)(x + 1)(x - 1), but these aren't considered "different" because 4 is a unit in Q.

in a ring with unity (which you have to have to even define units (invertible multiplicative elements)), any element r can ALWAYS be written r = u(u^{-1}r), for any unit u.

the definition of an irreducible in a ring R is something that cannot be written as the product of 2 non-units. that is:

u in R is irreducible if u = ab with a,b in R, implies a or b is a unit.

prime elements are irreducible, but it is not always true that irreducible elements are prime. for example: 3 in Z[√(-5)].