Algebraic Integers - Rotman - Proposition 2.70 - pages 118 - 119

[Math Amateur]

I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition).

I am currently focussed on Proposition 2.70 [pages 118 - 119] concerning algebraic integers.

I need help to the proof of part (iii) this Proposition.

Proposition 2.70 and its proof read as follows:View attachment 2709
View attachment 2710I have two questions pertaining to the proof of part (iii).

Question 1

In the above text when he is proving part (iii) Rotman writes the following:

" ... ... ... Now \(\displaystyle \mathbb{Z} [ \alpha \beta ] \) is an additive subgroup of \(\displaystyle G = \ < \alpha^i \beta^j \ : \ 0 \leq i \lt n \ , \ 0 \leq j \lt m > ... ... ... \)".

Now in part (ii) of the proof, Rotman has shown that:

" ... ... ... if \(\displaystyle deg(f) = n \) then \(\displaystyle \mathbb{Z} [ \alpha ] = G \) where G is the set of all set of all linear combinations \(\displaystyle m_0 + m_1 \alpha + \ ... \ ... + m_{n-1} \alpha^{n-1} \) ... ... "

So following this proof in part (ii) wouldn't we have, in the section of part (iii) quoted above, that \(\displaystyle \mathbb{Z} [ \alpha \beta ] = G \) where \(\displaystyle G = \ < \alpha^i \beta^j \ : \ 0 \leq i \lt n \ , \ 0 \leq j \lt m > ... ... ... \)" ... but Rotman calls \(\displaystyle \mathbb{Z} [ \alpha \beta ] = G \) a subgroup of G.

Can someone please clarify this issue?
Question 2

In the proof of part (iii) Rotman writes:

" ... ... ... Similarly, \(\displaystyle \mathbb{Z} [ \alpha + \beta ] \)is an additive subgroup of \(\displaystyle < \alpha^i \beta^j \ : \ i + j \leq n + m - 1 > \) and so \(\displaystyle \alpha + \beta \) is also an algebraic integer. ... ..."

Can someone please explain how this statement follows?

Help will be appreciated!

Peter

 

[Deveno]

The answer to your first question is: it's just poor writing on Mr. Rotman's part, they are different $G$'s (he should have used another letter for one or the other).

For the second question, it should be clear that:

$\Bbb Z[\alpha + \beta] \subseteq \langle \alpha^i\beta^j: i,j \in \Bbb N\rangle$

so the whole point is showing we only need finitely many of these products as a basis (considering $\Bbb Z[\alpha + \beta]$ as a $\Bbb Z$-module (an abelian group)).

There must be some earlier result about extensions, because the explanation given here is fairly terse.

One can show that if $f(\alpha) = 0$ and $\alpha_1 = \alpha,\alpha_2,\dots,\alpha_n$ are the roots of $f$, and similarly:

$\beta_1 = \beta,\beta_2,\dots,\beta_m$ are the roots of the monic $g$ for which $g(\beta) = 0$, that $\alpha + \beta$ is a root of:

$$F(x) = \prod_{i = 1}^n\prod_{j = 1}^m (x - (a_i + \beta_j))$$

which is a monic polynomial in $\Bbb Z[x]$.

The reasoning here is that $F$ is symmetric in the $\alpha_i$'s and $\beta_j$'s and so can be re-cast as a polynomial in the elementary symmetric polynomials of the $\alpha_i$ and $\beta_j$, and these elementary symmetric polynomials are the coefficients of $f$ and $g$, respectively (that is: they are integers).

Alternatively, one can show that $\Bbb Z[\alpha + \beta] \subseteq \Bbb Z[\alpha,\beta]$ which has the finite basis:

$\{ a^i\beta^j :0 \leq < i < n,0 \leq j < m\}$

by considering $\Bbb Z[\alpha,\beta]$ as a $\Bbb Z[\alpha]$-module.

It seems to me a lot of this would make more sense to you if you first studied the special case where you studied algebraic extensions of a field.

 

[Math Amateur]

The answer to your first question is: it's just poor writing on Mr. Rotman's part, they are different $G$'s (he should have used another letter for one or the other).

For the second question, it should be clear that:

$\Bbb Z[\alpha + \beta] \subseteq \langle \alpha^i\beta^j: i,j \in \Bbb N\rangle$

so the whole point is showing we only need finitely many of these products as a basis (considering $\Bbb Z[\alpha + \beta]$ as a $\Bbb Z$-module (an abelian group)).

There must be some earlier result about extensions, because the explanation given here is fairly terse.

One can show that if $f(\alpha) = 0$ and $\alpha_1 = \alpha,\alpha_2,\dots,\alpha_n$ are the roots of $f$, and similarly:

$\beta_1 = \beta,\beta_2,\dots,\beta_m$ are the roots of the monic $g$ for which $g(\beta) = 0$, that $\alpha + \beta$ is a root of:

$$F(x) = \prod_{i = 1}^n\prod_{j = 1}^m (x - (a_i + \beta_j))$$

which is a monic polynomial in $\Bbb Z[x]$.

The reasoning here is that $F$ is symmetric in the $\alpha_i$'s and $\beta_j$'s and so can be re-cast as a polynomial in the elementary symmetric polynomials of the $\alpha_i$ and $\beta_j$, and these elementary symmetric polynomials are the coefficients of $f$ and $g$, respectively (that is: they are integers).

Alternatively, one can show that $\Bbb Z[\alpha + \beta] \subseteq \Bbb Z[\alpha,\beta]$ which has the finite basis:

$\{ a^i\beta^j :0 \leq < i < n,0 \leq j < m\}$

by considering $\Bbb Z[\alpha,\beta]$ as a $\Bbb Z[\alpha]$-module.

It seems to me a lot of this would make more sense to you if you first studied the special case where you studied algebraic extensions of a field.

Thanks for the help and advice Deveno ... ...

Working through the details of your post now ... ...

Yes, agree ... I do need to revise field theory in general, I think, since I worked through it really quickly (to get an overview) and did not spend time on the details of the proofs of the theorems ... I think I also need to work some more examples to get a sense of the theory ... but I will do as you suggest and first revise algebraic extensions of a field as it seems to be intruding into ring and module theory ...

Thanks again for your help and support ... ...

Peter

 

[Math Amateur]

The answer to your first question is: it's just poor writing on Mr. Rotman's part, they are different $G$'s (he should have used another letter for one or the other).

For the second question, it should be clear that:

$\Bbb Z[\alpha + \beta] \subseteq \langle \alpha^i\beta^j: i,j \in \Bbb N\rangle$

so the whole point is showing we only need finitely many of these products as a basis (considering $\Bbb Z[\alpha + \beta]$ as a $\Bbb Z$-module (an abelian group)).

There must be some earlier result about extensions, because the explanation given here is fairly terse.

One can show that if $f(\alpha) = 0$ and $\alpha_1 = \alpha,\alpha_2,\dots,\alpha_n$ are the roots of $f$, and similarly:

$\beta_1 = \beta,\beta_2,\dots,\beta_m$ are the roots of the monic $g$ for which $g(\beta) = 0$, that $\alpha + \beta$ is a root of:

$$F(x) = \prod_{i = 1}^n\prod_{j = 1}^m (x - (a_i + \beta_j))$$

which is a monic polynomial in $\Bbb Z[x]$.

The reasoning here is that $F$ is symmetric in the $\alpha_i$'s and $\beta_j$'s and so can be re-cast as a polynomial in the elementary symmetric polynomials of the $\alpha_i$ and $\beta_j$, and these elementary symmetric polynomials are the coefficients of $f$ and $g$, respectively (that is: they are integers).

Alternatively, one can show that $\Bbb Z[\alpha + \beta] \subseteq \Bbb Z[\alpha,\beta]$ which has the finite basis:

$\{ a^i\beta^j :0 \leq < i < n,0 \leq j < m\}$

by considering $\Bbb Z[\alpha,\beta]$ as a $\Bbb Z[\alpha]$-module.

It seems to me a lot of this would make more sense to you if you first studied the special case where you studied algebraic extensions of a field.

Hi Deveno,

I am having some trouble with both of your explanations for the second question - but this is very likely due to the fact that I am not yet up to speed in the areas you draw upon to achieve the required result.

I note that your first explanation seems to draw on results that I am unaware of regarding symmetric polynomials ... I will have a search through some texts for such results ... ...

The understanding of your second explanation is probably where I think you are saying that I need to revise and extend my knowledge of field extensions.

Specifically, at my present state of knowledge, I am not aware of how \(\displaystyle \mathbb{Z} [ \alpha, \beta ] \) is defined and why it is true that \(\displaystyle \mathbb{Z} [ \alpha + \beta ] \subseteq \mathbb{Z} [ \alpha, \beta ] \).

Do we indeed define \(\displaystyle \mathbb{Z} [ \alpha, \beta ] \) as follows:

\(\displaystyle \mathbb{Z} [ \alpha, \beta ] = \{ g( \alpha ) , g( \beta ) \ | \ g(x) \in \mathbb{Z} (x) \}\) ... ...?

Maybe, as you say, I need to revisit algebraic extensions of a field ... ...

It is interesting to note that Rotman does not deal with field extensions until Section 2.9 Quotient Rings and Finite Fields on pages 160 - 171 ... which is a number of pages on from Proposition 2.70 (pages 118 - 119)

Peter

 

[Deveno]

In terms of complexity of structure, the following order is pretty much "standard":

Semigroups
Monoids
Commutative Monoids
Groups
Abelian Groups
Rings
Commutative Rings
Integral Domains
Unique Factorization Domains
Principal Ideal Domains
Euclidean Domains
Fields
Modules
Vector Spaces
Algebras

With semigroups (and thus with rings), it turns out that "less structure" means "greater variety", which is why there are several "steps" between rings and fields (and these steps are mirrored in some more exotic varieties of structures between semigroups and groups).

Some of these structures (such as vector spaces) have various "branchings" that come into play when we start to mix them with topological concepts, or more generally, various kinds of partial ordering lattices. The more closely these resemble the standard spatial structure of a real vector space, the more "analytic" (and less "algebraic") these structures become, and the underlying algebraic structure becomes less of a focal point for study.

So while in a sense it is perfectly reasonable to study rings first, and then treat fields as "special cases", in practice our INTUITIONS are better developed to study fields first, as they possesses most of the algebraic properties we are USED to.

With a set $S$, and a ring $R$, one typically denotes the RING one gets by minimally extending $R$ to include $S$ as $R$. If $S$ is a finite set, like $\{s_1,s_2\}$ it is common not to include the braces, so one writes $R[s_1,s_2]$. This process is called "adjunction". If $R$ and $S$ are part of some larger super-structure, the meaning of this is clear; if $S$ is not part of some "super-set" also containing $R$, then it's not so clear what $R$ ACTUALLY is; we distinguish such things as "formal $R$-linear combinations" (for example, when $R$ is commutative), and the astute reader will realize we are talking about some kind of "free object" here (that will possesses an appropriate universal property).

If $R = F$, a field, we are likewise interested in $F(S)$, which will typically (but not always) be somewhat larger than $R$ (because we need to have multiplicative inverses, as well). For the ring $F[x]$, the field $F(x)$, or rational functions in $x$, is created the same way we create the rational numbers from the integers, by forming its field of fractions.

So there is always this kind of "trade-off" between specific, and general, and also: difficult, and well-behaved. By the time you get a complete ordered archimedean field, the choices available are extremely limited (but we can do calculus!), whereas with an arbitrary ring, there are many viable examples (about which it is often hard to get concrete results).

Ideally, it's nice to find a balance: a concept which is general enough to have wide applicability, but specific enough to be easy to work with. Wide-angle, and zoom focus are the default settings, but this example here from Rotman is "somewhere in the middle".

 

[Math Amateur]

In terms of complexity of structure, the following order is pretty much "standard":

Semigroups
Monoids
Commutative Monoids
Groups
Abelian Groups
Rings
Commutative Rings
Integral Domains
Unique Factorization Domains
Principal Ideal Domains
Euclidean Domains
Fields
Modules
Vector Spaces
Algebras

With semigroups (and thus with rings), it turns out that "less structure" means "greater variety", which is why there are several "steps" between rings and fields (and these steps are mirrored in some more exotic varieties of structures between semigroups and groups).

Some of these structures (such as vector spaces) have various "branchings" that come into play when we start to mix them with topological concepts, or more generally, various kinds of partial ordering lattices. The more closely these resemble the standard spatial structure of a real vector space, the more "analytic" (and less "algebraic") these structures become, and the underlying algebraic structure becomes less of a focal point for study.

So while in a sense it is perfectly reasonable to study rings first, and then treat fields as "special cases", in practice our INTUITIONS are better developed to study fields first, as they possesses most of the algebraic properties we are USED to.

With a set $S$, and a ring $R$, one typically denotes the RING one gets by minimally extending $R$ to include $S$ as $R$. If $S$ is a finite set, like $\{s_1,s_2\}$ it is common not to include the braces, so one writes $R[s_1,s_2]$. This process is called "adjunction". If $R$ and $S$ are part of some larger super-structure, the meaning of this is clear; if $S$ is not part of some "super-set" also containing $R$, then it's not so clear what $R$ ACTUALLY is; we distinguish such things as "formal $R$-linear combinations" (for example, when $R$ is commutative), and the astute reader will realize we are talking about some kind of "free object" here (that will possesses an appropriate universal property).

If $R = F$, a field, we are likewise interested in $F(S)$, which will typically (but not always) be somewhat larger than $R$ (because we need to have multiplicative inverses, as well). For the ring $F[x]$, the field $F(x)$, or rational functions in $x$, is created the same way we create the rational numbers from the integers, by forming its field of fractions.

So there is always this kind of "trade-off" between specific, and general, and also: difficult, and well-behaved. By the time you get a complete ordered archimedean field, the choices available are extremely limited (but we can do calculus!), whereas with an arbitrary ring, there are many viable examples (about which it is often hard to get concrete results).

Ideally, it's nice to find a balance: a concept which is general enough to have wide applicability, but specific enough to be easy to work with. Wide-angle, and zoom focus are the default settings, but this example here from Rotman is "somewhere in the middle".

Thanks Deveno ... Just reading through and reflecting on what you have said ... Appreciate the overview as well as the guidance and help ...

Thanks again,

Peter

***EDIT*** Your order of structures surprised me (at first glance anyway ... Still reflecting on this matter ... ) ... I would have thought that fields would come after modules ... Not before ...