formal power series over field
Theorem. If K is a field, then the ring K[[X]] of formal power series is a discrete valuation ring with
(X) its unique maximal ideal.
Proof. We show first that an arbitrary ideal I of K[[X]] is a principal ideal. If
I=(0), the thing is ready. Therefore, let I≠(0). Take an element
f(X):=∞∑i=0aiXi |
of I such that it has the least possible amount of successive zero coefficients in its beginning; let its first non-zero coefficient be ak. Then
f(X)=Xk(ak+ak+1X+…). |
Here we have in the parentheses an invertible formal power series g(X), whence get the equation
Xk=f(X)[g(X)]-1 |
implying Xk∈I and consequently (Xk)⊆I.
For obtaining the reverse inclusion, suppose that
h(X):=bnXn+bn+1Xn+1+… |
is an arbitrary nonzero element of I where bn≠0. Because n≥k, we may write
h(X)=Xk(bnXn-k+bn+1Xn-k+1+…). |
This equation says that h(X)∈(Xk), whence I⊆(Xk).
Thus we have seen that I is the principal ideal (Xk), so that K[[X]] is a principal ideal domain.
Now, all ideals of the ring K[[X]] form apparently the strictly descending chain
(X)⊃(X2)⊃(X3)⊃…⊃(0), |
whence the ring has the unique maximal ideal (X). A principal ideal domain with only one maximal ideal is a discrete valuation ring.
Title | formal power series over field |
---|---|
Canonical name | FormalPowerSeriesOverField |
Date of creation | 2015-10-19 9:13:35 |
Last modified on | 2015-10-19 9:13:35 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 7 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 13H05 |
Classification | msc 13J05 |
Classification | msc 13F25 |