# 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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### Why characters are continuous

According to Wikipedia: ''Every character is automatically continuous from $A$  to $\mathbb C$, since the kernel of a character is a maximal ideal, which is closed. '' where $A$ is a Banach algebra. ...
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### Need some help finishing this proof about characters in Banach algebras

I tried to prove: Let $A$ be a commutative unital complex Banach algebra. Then there is a bijection between the maximal ideals in $A$ and the set of non-zero homomorphisms $A \to \mathbb C$. But I ...
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### Number of isomorphisms between two fields

Let $F,F'$ be two fields. Is there anything that can be said about the number of isomorphisms that can exist? In particular can there be more than one? What if $F$ is the complex numbers $\mathbb C$? ...
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### When does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$?

Let $p$ be a prime integer, and let $q=p^r$ and $q'=p^k$. For which values of $r$ and $k$ does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$? From Artin's Algebra, Chapter 15, problem 7.12 from the ...
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### Why $K(u)$ is a field?

Let $F$ be an extension field of $K$ and $u\in F$. How do we know that adjoining an element of F to K, makes $K(u)$ a field? I know that $Q(\sqrt2)=\{a+b\sqrt2|a,b\in Q\}$ is a field, but in the ...
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### Proof about finite dimensional vector spaces over fields

Prove that every finite dimensional vector space $V$of dimension $n$ over a field $F$ is isomorphic to the vector space $F^n$. Okay, lot's of stuff here. I think most of the reason I can not do this ...
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### Ring homomorphism with field as image, is the pre-image also a field?

Let $f:R\rightarrow S$ be a surjective ring homomorphism. $R,S$ are both integral domains. Suppose $S$ is a field, then is $R$ also a field? A possible useful fact: A finite integral domain is a ...
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### Analogy between the quaternion ring and extensions of the rationals

I've started studing fields and their extensions. As an exercise I proved that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$ by showing that $B=(1,\sqrt2,\sqrt3,\sqrt6)$ is a base for the extension field ...
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### Find primitive element of splitting field of $1 + x + x^2 - x^5$

As the title says, I need to find the primitive element of the splitting field of $1 + x + x^2 - x^5$ over $\mathbb{Q}$. Firstly, I would proceed by finding the roots as the splitting field has to ...
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### Ring theory question.

Why is a field with 27 elements has characteristic 3? I was solving a question and I came to know this fact which I didn't know before. Is there anyone who can explain this to me? Thanks in advance.
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### Constructing expressions

Suppose you are given all the elementary symmetric functions of $n$ variables $x_1,x_2,...,x_n$ and two rational functions $A(x_1,x_2,...,x_n)$ and $B(x_1,x_2,...,x_n)$ in the same $n$ variables that ...
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### Field extension of a finite field

Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements. I am not sure how to do this ...
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### Group-Isomorphism problem

I want to find an group-isomorphism $$\psi : (\mathbb{Z}/8\mathbb{Z},+) \longrightarrow \mathbb{F}_9^\times$$ which should be used to multiply elements in $\mathbb{F}_9$ or to find the inverse ...
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### Extension Fields and Quotients

In the Dummit and Foote 3ed chapter on field extensions (ch. 13), it is stated as a theorem (6) that $F(\alpha) \cong F[x]/(p(x))$ where $\alpha$ is a root of $p(x)$ and goes on to state that any ...
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### Field extensions and algebraic/transcendental elements

Let $E$ be an extension of field $F$, and let $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over $F$ but algebraic over $F(\beta)$. Show that $\beta$ is algebraic over $F(\alpha)$. Okay,...
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### Does $x\cdot 0 = 0$ follow from the field axioms alone?

From the field axioms alone, does it follow that $x \cdot 0 = 0$ for all $x$? All I would like is a statement that it can or cannot be done (hints not necessary). I would like to do it myself; I ...
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### Question About Notation In Field Theory.

I have a question about notation specifically square brackets $[$ and round brackets $($. My textbook doesn't explain any of this and I cannot find a reliable source online to confirm the difference. ...
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### Constructing splitting fields

Construct the splitting field for the polynomial $(x^4 - x^2 - 2)$ over $\mathbb Q$ (rationals) . What is the degree of the extension? Why? How would one go about tackling this question? I'm a bit ...
### The fixed field of a galois group if the characteristic is $p$.
Consider the following proposition with its relative proof: Let $k$ be an algebraically closed field of characteristic $0$. a) If $L$ is a subfield of $k$, then every elements of \$\operatorname {...