# 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 ...

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### Set of Functions is a Vector Space problem

Let $F$ be a ﬁeld. Consider the $F$-vector space $F(X, F)$ of all functions from $X = \{ 1, 2 \} \to F$. Deﬁne $e_1, e_2 \in F(X, F)$ by $e_1 = \{ (1, 1),(2, 0) \}$ and $e_2 = \{ (1, 0),(2, 1) \}$. ...
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### Applications of additive version of Hilbert's theorem 90

Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$, and $a$ is element of $L$ with trace 0, there exists an ...
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### $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ ??

Let $p,q$ be primes, $p≠q$, then I have to show that $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ So far I've tried a lot of things with minimal polynomials and bases, ...
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### Understanding a Proof in Galois Theory

The following is an extract from my Galois Theory course lecture notes. I understand the proof in the reverse direction so have included only the part of the proof that confuses me, even though it ...
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### Question regarding invariant separable elements of an algebraic closure of a field.

Let $\bar k$ be an algebraic closure of $k$ and $\alpha \in \bar k$ be separable over $k$. Suppose that $\sigma(\alpha)=\alpha$ for any non-zero ring homomorphism $\sigma:\bar k\rightarrow \bar k$. ...
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### If $\sigma : k \rightarrow L$ is a non-zero ring homomorphism of fields, show that $\sigma$ can be extended to an isomorphic copy of L

If $\sigma : k \rightarrow L$ is a non-zero homomorphism of fields, show that there is an extension field $E$ of $k$ that is isomorphic to $L$ by an isomorphism that extends $\sigma.$
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### Is a field determined by its family of general linear groups?

Assume that $K,L$ are fields such that there is an isomorphism of groups $\mathrm{GL}_n(K) \cong \mathrm{GL}_n(L)$ for all $n \in \mathbb{N}$. Does it follow that $K \cong L$? I am also interested in ...
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### $\operatorname{Gal}({\mathbb Q(i,\sqrt[4]{2})}/{\mathbb Q})$

I'm studying $\operatorname{Gal}({\mathbb Q(i,\sqrt[4]{2})}\Big/{\mathbb Q})$; I find it has order $8$. Any non-trivial subgroup will have order $2$ or $4$. Subgroups of order $2$ are easy to find, ...
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### Smallest subfield containing $F$ and $\alpha$ [duplicate]

Let $F$ be a field and let $K$ be an extension of $F$. Show that if $\alpha\in K$ is algebraic over $F$, $F[\alpha]=\{p(\alpha)\mid p(x)\in F[x]\}$ is the smallest subfield of $K$ containing $F$ and ...
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### About isomorphism of rings and fields

If $A,B$ are rings and $A$ is a field. If $A$ is a field and $A\cong B$ so $B$ is a field too? Thank you!
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### composition of field extensions

Let fields $K\subseteq L\subseteq M$. Then we know that if $L$ is a finite extension of $K$ and $M$ a finite extension of $L$, then $M$ is a finite extension of $K$. Can we generalize this property? ...
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### Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we ...
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### Proving $f = x^8+x^7+x^3+x+1$ to be irreducible over $\mathbf{F}_2$.

I am taking all the irreducible polynomials over $\mathbf{F}_2$ of degree 1 to 4 inclusive and using the long division to determine whether the given $f$ is divisible by anyone of them and it is not ...
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### Finding the order of an irreducible polynomial $f$ in $F_3[x]$ of degree 4?

The technique I am using is based on the long division of $x^e - 1$ (e is to be the order) which is really tiresome. So what the other methods (efficient)?
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### $[\mathbb{Q}(\alpha, \beta):\mathbb{Q}(\alpha)]\mid [\mathbb{Q}(\beta):\mathbb{Q}]$?

Given $\alpha, \beta$ algebraic numbers over $\mathbb{Q}$, it is known that $d=[\mathbb{Q}(\alpha, \beta):\mathbb{Q}(\alpha)]\le[\mathbb{Q}(\beta):\mathbb{Q}]=b$. It is also true that $d\mid b$ ? If ...
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### One question about additive identity arising from Apostol's field axiom in the Mathematical Analysis 2nd edition

In the begining of this book, the field axiom 4 talks about "Given any two real numbers x and y, there exists a real number z such that x+z=y and this z is denoted by y-x." Therefore, for each x, we ...
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### Example of a non-Galois extension with $[L : K] = |Aut(L/K)|$

When an extension of fields $L/K$ is finite, we always have $|\operatorname{Aut}(L/K)| \leq [L : K]$, and if $L/K$ is Galois then $|\operatorname{Aut}(L/K)| = [L : K]$. Is the converse true? Is ...
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### prove that the Galois group $Gal(L:K)$ is cyclic

Let $L$ be a field extension of a field $K$. Suppose $L=K(a)$, and $a^n\in K$ for some integer $n$. If $L$ is Galois extension of $K$ (i.e., $L$ is the splitting field of $f(x)\in K[x]$, and $f$ is ...
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### Proving integrality of the coefficients “inside the box”

Consider the (usual) $ABKL$ setting: $A$ is an integral domain with field of fractions $K$, $L/K$ is an algebraic field extension, and $B$ is the integral closure of $A$ in $L$ (we are not assuming ...
### Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements
I would like to know if my proof below is correct. Problem Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements. Solution If $\mathbb{F}$ is a ...
This is taken from the book Algebra by Thomas W. Hungerford ; Theorem. Let $K$ be an extension of $F$. The following are equivalent: $K$ is algebraic and Galois over $F$. $K$ is separable over $F$ ...