Galois theory (beginner question about normal algebraic extension: Supposedly Z2(X3)⊂Z2(X)Z2(X3)⊂Z2(X) is not a normal alg extension.)

MathNugget

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Supposedly Z2(X3)Z2(X)Z_2(X^3) \subset Z_2(X) is not a normal alg extension. I know by 1 definition that means if an irreducible polynomial from first field has a root in the 2nd, then it has all the roots in the 2nd.
Would x31=x3+1=(x1)(x2+x+1)=(x+1)(x2+x+1)x^3-1 = x^3+1=(x-1)(x^2+x+1)= (x+1)(x^2+x+1) solve this? the last polynomial doesn't have any root in Z2(X)Z_2(X), so it would be an example to prove that is not a normal extension...
 
Supposedly Z2(X3)Z2(X)Z_2(X^3) \subset Z_2(X) is not a normal alg extension. I know by 1 definition that means if an irreducible polynomial from first field has a root in the 2nd, then it has all the roots in the 2nd.
Would x31=x3+1=(x1)(x2+x+1)=(x+1)(x2[SIZE=6]+[/SIZE]x+1)x^3-1 = x^3+1=(x-1)(x^2+x+1)= (x+1)(x^2[SIZE=6]+[/SIZE]x+1) solve this? the last polynomial doesn't have any root in Z2(X)Z_2(X), so it would be an example to prove that is not a normal extension...
a^3 - b^3 = (a - b)(a^2 + a*b + b^2) ....... and.........

a^3 + b^3 = (a + b)(a^2 - a*b + b^2)

Check your work carefully....
 
I am assuming that Z2\mathbb Z_2 is the field with two elements (0,1), but what do Z2(X3)\mathbb Z_2(X^3) and Z2(X)\mathbb Z_2(X) mean ?
 
a^3 - b^3 = (a - b)(a^2 + a*b + b^2) ....... and.........

a^3 + b^3 = (a + b)(a^2 - a*b + b^2)

Check your work carefully....
True, but since it's Z2\mathbb{Z}_2, I figured -1 = +1 . Or maybe I misunderstood your comment? I guess I didn't write that properly, it wasn't a random Z, it was Z\mathbb{Z} (integers).
I am assuming that Z2\mathbb Z_2 is the field with two elements (0,1), but what do Z2(X3)\mathbb Z_2(X^3) and Z2(X)\mathbb Z_2(X) mean ?
Your guess is as good as mine, since I cannot seem to find this notation somewhere else. Given the inclusion, let's say:
Z(X)=anXn+...+a1X+a0Z(X3)=anX3n+...+a1X3+a0\mathbb{Z}(X)=a_nX^n+...+a_1X+a_0 \Rightarrow \mathbb{Z}(X^3)=a_nX^{3n}+...+a_1X^3+a_0 .
 
But then Z2(X)\mathbb Z_2(X) and Z2(X3)\mathbb Z_2(X^3) are just rings, not fields.
 
But then Z2(X)\mathbb Z_2(X) and Z2(X3)\mathbb Z_2(X^3) are just rings, not fields.
You're right. I realize now, nowhere in the material I read did they mention that normal algebraic extensions refer only to fields.
Therefore, I am trying to prove it's a normal algebraic ring extension... with the same definitions and all that.
 
a^3 - b^3 = (a - b)(a^2 + a*b + b^2) ....... and.........

a^3 + b^3 = (a + b)(a^2 - a*b + b^2)

Check your work carefully....
There is no difference between "+" and "-" in Z2\mathbb Z_2.
 
Supposedly Z2(X3)Z2(X)Z_2(X^3) \subset Z_2(X) is not a normal alg extension. I know by 1 definition that means if an irreducible polynomial from first field has a root in the 2nd, then it has all the roots in the 2nd.
Would x31=x3+1=(x1)(x2+x+1)=(x+1)(x2+x+1)x^3-1 = x^3+1=(x-1)(x^2+x+1)= (x+1)(x^2+x+1) solve this? the last polynomial doesn't have any root in Z2(X)Z_2(X), so it would be an example to prove that is not a normal extension...
Isn't 1 a root of x3±1x^3\pm 1 in Z2\mathbb Z_2 ?

More general question: what are you trying to solve or prove?
 
Isn't 1 a root of x3±1x^3\pm 1 in Z2\mathbb Z_2 ?

More general question: what are you trying to solve or prove?
I am trying to prove that Z2(X3)Z2(X)\mathbb{Z}_2(X^3) \subset \mathbb{Z}_2(X) is a normal algebraic extension. One definition would be "all irreducible polynomials from the first set (polynomials from that ring/field) that have at least 1 root in the bigger set (here Z2(X)\mathbb{Z}_2(X) ), have all roots in that bigger set."

So I am on the hunt for a polynomial from Z2(X3)\mathbb{Z}_2(X^3) with some roots, but not all, in Z2(X)\mathbb{Z}_2(X).

I think X3+1X^3 + 1 , as you said, has 1 root in the 2nd ring ( x = 1), but the other 2 roots are obviously not in the ring. I guess I am more interested in whether or not I completely misunderstood what a normal extension is, or the solution was this simple / obvious...
 
I am trying to prove that Z2(X3)Z2(X)\mathbb{Z}_2(X^3) \subset \mathbb{Z}_2(X) is a normal algebraic extension. One definition would be "all irreducible polynomials from the first set (polynomials from that ring/field) that have at least 1 root in the bigger set (here Z2(X)\mathbb{Z}_2(X) ), have all roots in that bigger set."

So I am on the hunt for a polynomial from Z2(X3)\mathbb{Z}_2(X^3) with some roots, but not all, in Z2(X)\mathbb{Z}_2(X).

I think X3+1X^3 + 1 , as you said, has 1 root in the 2nd ring ( x = 1), but the other 2 roots are obviously not in the ring. I guess I am more interested in whether or not I completely misunderstood what a normal extension is, or the solution was this simple / obvious...
But x3±1x^3\pm 1 is not irreducible since it has a root (=1) in Z2\mathbb Z_2.

I am not familiar with normal algebraic extensions of rings, and my online search wasn't too successful. Is this a problem from your homework or a book you study ? Can you post an exact statement, or a snapshot, of the problem?

Thank you.
 
But x3±1x^3\pm 1 is not irreducible since it has a root (=1) in Z2\mathbb Z_2.

I am not familiar with normal algebraic extensions of rings, and my online search wasn't too successful. Is this a problem from your homework or a book you study ? Can you post an exact statement, or a snapshot, of the problem?

Thank you.
Sadly it's not from a book. The definitions I got are:
kKk \subset K is an algebraic extension. We call it normal if σ:KKˉ\forall \sigma : K \rightarrow \bar K a k - morphism σ\rightarrow \sigma k-automorphism of K.
Kˉ\bar K is the algebraic closure of K.

equivalently:
i) f(x)k(x)\forall f(x) \in k(x) irreducible (in k), if it has a root in K, it has all roots in K.
There's also a 3rd definition, I have trouble translating.

The wikipedia article seems right on it, though, with the 3 equivalent definitions:
 
I'll write here the definitions from wikipedia:
"Let L/K be an algebraic extension (i.e., L is an algebraic extension of K), such that LKˉL \subseteq \bar K (i.e., L is contained in an algebraic closure of K). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:
1) Every embedding of L in Kˉ\bar K over K induces an automorphism of L.
2) L is the splitting field of a family of polynomials in K[X]K[X].
3) Every irreducible polynomial of K[X]K[X] that has a root in L splits into linear factors in L. "

It's a homework, by the way.
 
I'll write here the definitions from wikipedia:
"Let L/K be an algebraic extension (i.e., L is an algebraic extension of K), such that LKˉL \subseteq \bar K (i.e., L is contained in an algebraic closure of K). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:
1) Every embedding of L in Kˉ\bar K over K induces an automorphism of L.
2) L is the splitting field of a family of polynomials in K[X]K[X].
3) Every irreducible polynomial of K[X]K[X] that has a root in L splits into linear factors in L. "

It's a homework, by the way.
All the definitions so far are for fields. As for rings, I've found a couple of mentions of ring extensions on Wikipedia, but none mentions normal extensions:
  1. https://en.wikipedia.org/wiki/Algebra_extension
  2. https://en.wikipedia.org/wiki/Subring#Ring_extensions
Do you want to post the exact copy of the homework ?
 
All the definitions so far are for fields. As for rings, I've found a couple of mentions of ring extensions on Wikipedia, but none mentions normal extensions:
  1. https://en.wikipedia.org/wiki/Algebra_extension
  2. https://en.wikipedia.org/wiki/Subring#Ring_extensions
Do you want to post the exact copy of the homework ?
Sure, but it's not from a book. The whole thing is:
Exercise: Z2(X3)Z2(X)\mathbb{Z}_2(X^3) \subset \mathbb{Z}_2(X) is not normal...
I doubt it's about normal subgroups, given the next exercise is
Q(X2X3+1)Q(X)\mathbb{Q}(\frac{X^2}{X^3+1}) \subset \mathbb{Q}(X) is not normal.

I suppose the notion for fields can be directly applied to rings. All the properties above can be checked for rings...
 
I doubt it's about normal subgroups, given the next exercise is
I am really rusty on these subjects (not too surprising after not dealing with it for about half-century :)): the next exercise reminded me that F(X)F(X) is typically used, at least in my textbooks, for fields of rational polynomial functions, as opposed to F[X]F[X] used to denote polynomial rings. Sorry about a whole bunch of misleading posts :(

At this point I have no clue how to solve this kind of problems (sorry again:(), but will appreciate if you post the answers if/when you get them.
 
I am really rusty on these subjects (not too surprising after not dealing with it for about half-century :)): the next exercise reminded me that F(X)F(X) is typically used, at least in my textbooks, for fields of rational polynomial functions, as opposed to F[X]F[X] used to denote polynomial rings. Sorry about a whole bunch of misleading posts :(

At this point I have no clue how to solve this kind of problems (sorry again:(), but will appreciate if you post the answers if/when you get them.
I understand what you're saying. Teachers at this uni have a habit of changing notations often (as 1 of them says, "to further confuse the students").
I'll try to find a solution to the first question, although I think it's the one I posted above.

For the second one though, I am not even convinced
Q(X2X3+1)Q(X)\mathbb{Q}(\frac{X^2}{X^3+1}) \subset \mathbb{Q}(X) is true (the inclusion). But maybe this one too is as simple as taking a polynomial in first set, considering it's degree = t, and amplifying the polynomial with (X3+1)t(X^3+1)^t ...

Also, don't worry about it. I am messing up the notations and proper words to describe most of the things I have questions about. You've helped me a lot in the past months :)
 
Just a thought: polynomial u3x3=0u^3 - x^3 = 0 (whose both coefficients are in Z2(X3)\mathbb Z_2(X^3)) has a root u=xu=x, so uZ2(X)u\in \mathbb Z_2(X) but uZ2(X3)u\notin \mathbb Z_2(X^3). I believe it can be shown that u2+ux+x2u^2 + ux + x^2 has no roots in Z2(X)\mathbb Z_2(X) -- can you prove this? Do you agree this would prove "abnormality" of this extension?
 
Just a thought: polynomial u3x3=0u^3 - x^3 = 0 (whose both coefficients are in Z2(X3)\mathbb Z_2(X^3)) has a root u=xu=x, so uZ2(X)u\in \mathbb Z_2(X) but uZ2(X3)u\notin \mathbb Z_2(X^3). I believe it can be shown that u2+ux+x2u^2 + ux + x^2 has no roots in Z2(X)\mathbb Z_2(X) -- can you prove this? Do you agree this would prove "abnormality" of this extension?
Absolutely. Given that it's Z2\mathbb{Z}_2, though, proving it for a general polynomial u3x3=0u^3 - x^3 = 0 seems rather redundant (we're just putting x = 1, x = 0, getting 2 conditions: u20andu2+u+10u^2 \neq 0 \: and \: u^2 + u + 1 \neq 0 . it all ends up back to finding u = 1.

For u = 0, we see the example doesn't really work. And I do think the example for u = 1 would fit right into the 3rd definition of normal extension :).
 
Absolutely. Given that it's Z2\mathbb{Z}_2, though, proving it for a general polynomial u3x3=0u^3 - x^3 = 0 seems rather redundant (we're just putting x = 1, x = 0, getting 2 conditions: u20andu2+u+10u^2 \neq 0 \: and \: u^2 + u + 1 \neq 0 . it all ends up back to finding u = 1.

For u = 0, we see the example doesn't really work. And I do think the example for u = 1 would fit right into the 3rd definition of normal extension :).
Not sure I follow this, but would be interesting to know what your professor thinks. u=1Z2(X3)u=1 \in \mathbb Z_2(X^3), unlike the case of u=xu=x.
 
Not sure I follow this, but would be interesting to know what your professor thinks. u=1Z2(X3)u=1 \in \mathbb Z_2(X^3), unlike the case of u=xu=x.
True, but as far as I know, f is reducible if there's g and h in the same ring/field, and f = gh.
Here, 1 is root, but x31=(x1)(x2+x+1)x^3 - 1 = (x-1) (x^2 + x + 1), and x1,x2+x+1Z2(X3)x-1, \: x^2 + x + 1 \notin \mathbb{Z}_2(X^3).

I submitted the homework to the teacher (at least what I did from it), but I don't expect an answer (not even a grade), so it would be hard to guess whether it's correct or wrong...
 
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