# 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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### What are all the Algebraic Elements of $F(t)$ over $F$?

Let $F$ be a field and $t$ be a variable. Let $F(t)$ be the field of quotients of the polynomial ring $F[x]$. Question. What are all the elements in $F(t)$ which are algebraic over $F$? I think ...
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### Show that $\ker(\phi)$ is a maximal ideal if and only if $B$ is a field

Let $A$ and $B$ two commutative rings with unity $1_A \not= 0_A$ and $1_B \not= 0_B$. Consider $\phi : A \to B$ a ring epimorphism. Show that if $\ker(\phi)$ is a maximal ideal, $B$ is a field. I ...
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### Can this extension of fields be transcendental?

Let $(R, \mathfrak m)$ be a local integral domain which is contained in a field $K$. Let $0 \neq x \in K$ be such that $\mathfrak m R[x]$ is a proper ideal of $R[x]$ (one can show for any $x$ that ...
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### Is Z3 a sub-field of R?

The inverse numbers for the items in $\mathbb{Z3}$ are different than in $\mathbb{R}$ so I assume it's not a sub-field of $\mathbb{R}$. Am I correct? And in general, can sub-field of a infinite ...
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### Product of two transcendental numbers is transcendental

let $\alpha,\beta$ be transcendental numbers.which of the following are true? 1)$\alpha\beta\ \text{ is transcendental}$. 2)$\mathbb{Q}(\alpha)\ \text{is isomorphic to }\mathbb{Q}(\beta)$ ...
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### To find a field of $p^{2}$ elements ,where $p$ is prime

Show that there exists a finite field of $p^{2}$ elements for every prime $p\in\mathbb{N}$. What I thought is that if I find some irreducible polynomial of degree two over $\mathbb{Z_p[x]}$, then I ...
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### Adjoining a separable element to a field makes the extension separable

Let $k$ be a field, and $K$ be an extension. Suppose $a\in K$ is separable over $k$. What's a clever way to show that $k(a)$ is separable (i.e., that all elements of $k(a)$ have separable minimal ...
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### Norm on “tower” of fields (the question comes from algebraic number theory)

Consider the field extensions $L\supset K'\supset K$, where both $L|K$ and $K'|K$ are finite and Galois. I want to prove that $$N_{L|K}(L^\ast)\subseteq N_{L|K'}(L^{\ast})$$ Maybe it is very easy, ...
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### Isomorphism from $\mathbb{Q}(\sqrt{2})$ to $\mathbb{Q}[x]/\langle x^2 - 2\rangle$ [on hold]

I am just now beginning my first course in Fields. Sometimes I learn best by just being absolutely certain of some basic facts. This is why I like to ask simple True/False questions that I think are ...
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### Finite Galois Field extension of a field $F$ containing all roots of unity

Let $F$ be a field that contains all roots of unity. Furthermore, let $K$ be a finite algebraic extension of $F$ with abelian Galois group . Then $$K= F(z_1,\ldots , z_n)$$ for some $z_i \in K$ ...
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### Formally real field with two different orders

If $F$ is a formally real field, then there exists a total order relation on $F$ which is compatible with its sum and product, but it needs not be unique. a) How different can be two (compatible with ...