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I try to determine all the maximal ideals of the domain $\mathbb F_2+T\mathbb F_4[[T]]$. I even do not know whether it is a Dedekind domain. In case it is not, can one determine its prime ideals too?

Thanks in advance for any hints or solutions.

Bernard
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joaopa
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  • I'm not sure about the answer to this question, but a place to start might be to write everything over $\mathbb F_2$. So for example, you can think of $\mathbb F_4$ as the $\mathbb F_2$-algebra $\mathbb F_2 + \omega \mathbb F_2$ where $\omega^2 + 1 + \omega$. But careful, this then induces a different multiplication operation on the elements with $\text{deg}(T) \geq 1$. From there you can think about it as $\mathbb F_2[[T]] + \omega T \mathbb F_2[[T]]$. – dbossaller Oct 08 '21 at 15:52
  • Do you know how to classify the maximal ideals of $\mathbb{F}_4[[T]]$? If so, note that $\mathbb{F}_4[[T]]$ is a finite extension of your ring, so every maximal ideal of your ring lies under a maximal ideal of $\mathbb{F}_4[[T]]$... – diracdeltafunk Oct 08 '21 at 17:37

1 Answers1

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There is only one: $0 + T\mathbb F_4[[T]]$. Obviously this is a maximal ideal, as the quotient is $\mathbb F_2$, a field.

Now it's only necessary to show that $1+T\mathbb F_4[[T]]$ are all invertible, therefore they cannot be contained in any maximal ideal. This is essentially to compute the power series expansion $\dfrac{1}{1 + a_1T + a_2T^2 + a_3T^3+\cdots}$ which can be done inductively.

Indeed, this is a well-known fact: $\sum_{i=0}^\infty a_i T^i\in R[[T]]$ is invertible iff $a_0$ is invertible in $R$. See e.g. https://www.planetmath.org/invertibleformalpowerseries

Just a user
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