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Math Help - Formal Power Series

  1. #1
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    Formal Power Series

    Let  F be a field. Consider the ring  R = F[[t]] of formal power series in  t i.e. if  a \in F[[t]] , then  a=\sum_{n=0}^{\infty} a_nt^n where  a_n\in F .

    Multiplication is defined by  \left(\sum_{n=0}^{\infty} a_nt^n\right)\cdot\left(\sum_{n=0}^{\infty} b_nt^n\right) = \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} a_k b_{n-k} t^n\right) .

    (a) Prove that  \alpha \in R is a unit if and only if the constant term  a_0 \neq 0. (ex.  1-t is the inverse of  1+t+t^2+t^3+t^4+... )
    (b) Prove that  R is a Euclidean domain with respect to the norm  N(\alpha) = n if  a_n is the
    first term of  \alpha that is non-zero.
    (c) In the polynomial ring  R[x] , prove that  x^n-t is irreducible.
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  2. #2
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    Quote Originally Posted by mathman88 View Post
    Let  F be a field. Consider the ring  R = F[[t]] of formal power series in  t i.e. if  a \in F[[t]] , then  a=\sum_{n=0}^{\infty} a_nt^n where  a_n\in F .

    Multiplication is defined by  \left(\sum_{n=0}^{\infty} a_nt^n\right)\cdot\left(\sum_{n=0}^{\infty} b_nt^n\right) = \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} a_k b_{n-k} t^n\right) .

    (a) Prove that  \alpha \in R is a unit if and only if the constant term  a_0 \neq 0. (ex.  1-t is the inverse of  1+t+t^2+t^3+t^4+... )
    (b) Prove that  R is a Euclidean domain with respect to the norm  N(\alpha) = n if  a_n is the
    first term of  \alpha that is non-zero.
    (c) In the polynomial ring  R[x] , prove that  x^n-t is irreducible.

    Good. What've you tried so far and where are you stuck? Anyone seeing this stuff must be a medium undergraduate level in algebra, at least, so you must have some ideas. Let's see.

    Tonio
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  3. #3
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    Quote Originally Posted by mathman88 View Post
    Let  F be a field. Consider the ring  R = F[[t]] of formal power series in  t i.e. if  a \in F[[t]] , then  a=\sum_{n=0}^{\infty} a_nt^n where  a_n\in F .

    Multiplication is defined by  \left(\sum_{n=0}^{\infty} a_nt^n\right)\cdot\left(\sum_{n=0}^{\infty} b_nt^n\right) = \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} a_k b_{n-k} t^n\right) .

    (a) Prove that  \alpha \in R is a unit if and only if the constant term  a_0 \neq 0. (ex.  1-t is the inverse of  1+t+t^2+t^3+t^4+... )
    (b) Prove that  R is a Euclidean domain with respect to the norm  N(\alpha) = n if  a_n is the
    first term of  \alpha that is non-zero.
    (c) In the polynomial ring  R[x] , prove that  x^n-t is irreducible.
    (a): let \alpha=\sum_{n=0}^{\infty}a_nt^n. it's obvious that if a_0=0, then there exists no \beta \in F[[t]] such that \alpha \beta = 1 because the constant term of \alpha \beta is 0.

    if a_0 \neq 0, then a_0 is invertible in F. define b_0=a_0^{-1} and inductively b_n=-a_0^{-1}\sum_{j=1}^n a_jb_{n-j}, \ n \geq 1. let \beta=\sum_{n=0}^{\infty}b_nt^n and see that \alpha \beta = 1.

    (b) straightforward.

    (c) this part is nice! suppose x^n - t = r(x) s(x), for some r(x)=\sum_{i=0}^k r_ix^i, \ s(x)=\sum_{i=0}^m s_i x^i, \ k, m > 0, \ r_i, s_i \in F[[t]]. then r_0s_0=-t and

    hence, without loss of generality, we may assume that r_0=tu, \ s_0=-u^{-1}, for some invertible element u \in F[[t]]. to complete the proof of

    (c), use induction to show that, in F[[t]], we have t \mid r_i, for all i, which is the contradiction we need.
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  4. #4
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    Quote Originally Posted by NonCommAlg View Post
    (a): let
    (c) this part is nice! suppose x^n - t = r(x) s(x), for some r(x)=\sum_{i=0}^k r_ix^i, \ s(x)=\sum_{i=0}^m s_i x^i, \ k, m > 0, \ r_i, s_i \in F[[t]]. then r_0s_0=-t and

    hence, without loss of generality, we may assume that r_0=tu, \ s_0=-u^{-1}, for some invertible element u \in F[[t]]. to complete the proof of

    (c), use induction to show that, in F[[t]], we have t \mid r_i, for all i, which is the contradiction we need.
    Very nice!

    So the contradiction arises when we set  r(x) = t\cdot p(x) , so then  x^n-t = t\cdot p(x)\cdot s(x) \Longrightarrow t \mid x^n-t since no inverse of  t exists (shown in part a) to cancel out  t in the right hand side.  \longrightarrow\longleftarrow
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  5. #5
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    Quote Originally Posted by mathman88 View Post
    Very nice!

    So the contradiction arises when we set  r(x) = t\cdot p(x) , so then  x^n-t = t\cdot p(x)\cdot s(x) \Longrightarrow t \mid x^n-t since no inverse of  t exists (shown in part a) to cancel out  t in the right hand side.  \longrightarrow\longleftarrow
    correct!
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  6. #6
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    Tonio,

    Medium undergraduate?!? Im in the last of my undergraduate algebra classes and rings is what we just started with 3 weeks left so we must not be going that far because that makes no sense to me....what school did you go to?
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  7. #7
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    Quote Originally Posted by ChrisBickle View Post
    Tonio,

    Medium undergraduate?!? Im in the last of my undergraduate algebra classes and rings is what we just started with 3 weeks left so we must not be going that far because that makes no sense to me....what school did you go to?

    Hebrew University in Jerusalem. Indeed, we studied formal powers series in some algebra course in the second half part of undergraduate studies, and got deep into it in graduate courses. I wrote "at least" since imo there could be some school somewhere where this stuff is studied in 2nd-3rd year.
    Now basic ring theory we studied as a final chapter to linear algebra 1 and then again, more thorough, as part of Agebraic Structures in 2nd year, as preparation to Fields and Galois theory, so it could be they're going to teach you guys all this stuff in your last year...but it looks a little too little to me.
    My school though is considered as rather strong in algebra in general, with some emphasis in group theory, and perhaps this has something to do.

    Tonio
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  8. #8
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    Quote Originally Posted by NonCommAlg View Post
    (b) straightforward.
    What do you mean by that?
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  9. #9
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    Quote Originally Posted by Jim63 View Post
    What do you mean by that?
    well, it means "it's straightforward but i'm too lazy to write it down!" haha ... anyway, you need to check two conditions.

    1) N(\alpha \beta) \geq N(\alpha), for all 0 \neq \alpha, \beta \in F[[t]]: well, from the definition of N it's clear that N(\alpha \beta)=N(\alpha) + N(\beta). so this part is done.

    2) suppose that \alpha=\sum_{i=m}^{\infty}a_it^i, \ \beta=\sum_{i=k}^{\infty}b_it^i \in F[[t]], where k,m \geq 0, \ a_mb_k \neq 0. we want to prove that there exist \gamma, \delta \in F[[t]] such that \alpha=\gamma \beta + \delta and either \delta=0 or N(\delta) < N(\beta):

    if m=N(\alpha) < N(\beta)=k, then choose \gamma=0, \ \delta=\alpha. if m \geq k, then put \alpha=t^m \alpha_0, \ \beta=t^k\beta_0, where \alpha_0=a_m + a_{m+1}t + \cdots , \ \beta_0=b_k + b_{k+1}t + \cdots. now \beta_0 is invertible since b_k \neq 0.

    so there exists u \in F[[t]] such that \beta_0u=1. thus \alpha=t^{m-k}\alpha_0 u \beta. so in this case we can choose \delta=0, \ \gamma=t^{m-k} \alpha_0 u. \ \Box
    Last edited by NonCommAlg; February 14th 2010 at 11:51 PM.
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