# Tagged Questions

Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that topic....

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### Find the number of subfields of a field of cardinality $2^{100}$ [duplicate]

Find the number of subfields of a field of cardinality $2^{100}$ I want to know whether the answer is $9$. But I need a proper logic of that answer.
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### Irreducible polynomial of degree $5$ in $L[X]$ using Kummer's theory

Let $L=\mathbb{F_2}(\theta)$ where $\theta$ is a root of $X^4+X+1$ I am trying to solve the following consecutive questions in Galois Theory: Prove that $L$ has degree $4$ over $\mathbb{F_2}$ ...
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### Finite separable fields extensions and discriminant

I am supposed to prove that for a finite separable field extension $L/K$, the discriminant $Discr_{L/K}$ is not zero. (For a basis $\{a_1,\ldots,a_n\}$ of $L$ over $K$, the discriminant is defined by ...
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### Splitting field of $x^9-x$ over $\mathbb{Z}_3$.

Let $F$ be a splitting field of $x^9-x$ over $\mathbb{Z}_3$. $1$. Show that there is an $\alpha \in F - \mathbb{Z}_3$ such that $\alpha^2=2$. $2$. Show that $\mathbb{Z}_3(\alpha)$ is a splitting ...
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### minimal polynomial of $\alpha - \beta$ from the minimal polynomial of $\alpha$

If $f \in k[x]$ is the minimal polynomial of $\alpha \in K$, show how to write down the minimal polynomial of $\alpha - \beta$ for $\beta \in k$. I've been playing around with the minimal polynomial ...
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### $G$ has normal subgroup of order 5

Let $L$ be the splitting field of $x^5-7$ over $Q$ and let $G=\text{Gal}(L/Q)$ (I) Prove that $G$ has a normal subgroup of order $5$ (II) Prove that $G$ has a subgroup of order $4$ that is not ...
### Find $[\mathbb Q(\sqrt3, \sqrt{5},2^{\frac13}):Q]$
$[Q(\sqrt3, \sqrt5:Q]=4$  \begin{matrix} & & \mathbb Q(\sqrt3, \sqrt{5})(2^{\frac13}) & & \\ & \stackrel{a}{\diagup} & & \stackrel{b}{\diagdown} \\ \mathbb Q(\sqrt3, \...
### How can I prove that $\mathbb Z_3[i]\cong \mathbb Z_3[x]/\langle x^2+1\rangle$?
Question: Let $\mathbb Z_{3}\left [ i \right ]=\left \{ a+bi\mid a,b \in \mathbb{Z}_{3} \right \}.$ Show that the field $\mathbb Z_{3}\left [ i \right ]$ is ring isomorphic to the field \$\...