# Tag Info

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• 1,368

### A quantity associated to a field extension

For the rest of this post, I'm talking about fields not of characteristic $2$. Under this assumption, we will show that any such $V$ is either a subfield of $E$ or is a complement of a subfield $F'$ ...
• 4,991
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### Is the minimal polynomial of an algebraic formal Laurent series always separable?

Yes. Let $p=char(K)$ and $\alpha\in \overline{K(x)}\cap K((x))$ assumed to be inseparable over $K(x)$. Let $L= K^{1/p^\infty}$ which is perfect. If $\alpha$ is inseparable over $L(x)$ then $\alpha$'s ...
• 3,425

### Artin-Schreier theorem for rings (a little different)

I am assuming that you intended for $R$ to already be integrally closed in $K$, i.e. normal. Then the answer to your question is yes. Claim: Suppose the domain $R$ is integrally closed in its field of ...
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### A quantity associated to a field extension

For $F$ of characteristic $2$, a space $V$ has this form if and only if there are fields $F_1, F_2$ with $F \subseteq F_1 \subseteq F_2 \subseteq E$ and $F_2^2 \subseteq F_1$, and $V$ is an $F_1$-...
• 139k
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• 39.2k

### Counter example of a radical extension that is not Galois/normal over $\mathbb{Q}(\omega)$?

The polynomial $f:=(x^2-1)^3-2$ has Galois group $G \simeq S_4 \times C_2$ over $\mathbb{Q}$, so the commutator subgroup $[G,G] \simeq A_4$. Let $L$ be the splitting field (over $\mathbb{Q}$) of $f$. ...

### Complete reducibility and field extension

It is worth pointing out that there are general statements that semisimplicity can be checked over an algebraic closure. The following statement is used: Let $k$ be a field and $A$ a $k$-linear ...
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### Algorithms for Polynomials Over a Real Algebraic Number Field, a reference

The thesis can be found here.
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### General linear group action on extensions of finite fields

Let $F$ be any field and $F<E$ a finite field extension. Fix $x\in E^*$ and consider the multiplication operator $g_x\in \text{GL}_F(E)$, the group of invertible $F$-linear transformations of $E$. ...
• 11.5k
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• 23.6k
1 vote
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### Shrinking the base field of an affine variety

A standard reference is Chapter 6 of "Neron Models" of Bosch, Lütkebohmert and Raynaud, Springer, 1990. The original Grothendieck's exposés in the Cartan Seminar (1960--1961) are available ...
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• 12.1k
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### Extension of isomorphism of fields

In order to avoid having a question without answer in the forum, Eric Wofsey posted an answer here.
1 vote

### Field extensions (non-algebraic)

If $X$ is fibered over $C$ and $C$ has positive genus then, as $C$ has non-zero holomorphic differentials, you can pull back those to $X$, which is already a non-trivial condition on $X$. Moreover, ...
• 30.4k

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