# 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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### Show that $\mathbb{F}_9 \not \subset \mathbb{F}_{27}$

The usual answer will go like this: Since $2 \not | \ 3$ and $\mathbb{F}_{p^r} \subset \mathbb{F}_{p^s}$ if and only if $r | s$, then $\mathbb{F}_9 \not\subset \mathbb{F}_{27}$. However, I'm ...
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### Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
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### Mention a concrete example of a field (like $\mathbb{R}$) [closed]

This example must not be complete, I'm trying to understand the contrast between those two kind of fields, this is the problem, I can't even find an example. Thank you!.
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### Subfields of $\mathbb{C}$ which are connected with induced topology

The ring of continuous functions on $[0,1]$ to $\mathbb{R}$ has an interesting property: every maximal ideal of this ring is the subset of all functions vanishing at a common point. If we ...
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### prove that $x^{2\cdot 3^n}+x^{3^n}+1$ is not primitive (mod 2)

I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, It seems that there are no primitive polynomial of the form $$x^{2k}+x^{k}+1\quad k>1$$ and I'...
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### Minimal polynomial of $\alpha = \cos\left(\frac{\pi}{48}\right)$ over $\mathbb Q$

This is a homework problem, so just a nudge in the right direction would be great. So I am required to show that $\alpha$ is a algebraic over $\mathbb Q$ and show that the degree of its minimal ...
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### If $F$ is a field, what does the notation $F(x)$ mean?

If $F$ is a field, what does the notation $F(x)$ mean? I am trying to understand transcendence degree of field extension, and I am stuck in this notation. More context: I am reading this pdf, and my ...
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### Finding the Galois group of $x^4+5x^2+5$

Find the Galois group of $f(x)=x^4+5x^2+5\in \mathbb{Q}[x]$. This is solved here, Exersice 3: https://math.berkeley.edu/~serganov/114/solhwg.pdf I have a question about it (I will not write all the ...
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### Points with power distances

It's possible for seven points to be at integral distances. I'm dallying with powered triangles, though, so I'm looking for point sets where all distances are powers of a fixed $x$ value. For example,...
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### Considerations on cyclotomic extensions

I stopped in front of some issues regarding certain passages of the theorem 9.4 of Algebra of Serge Lang, in which we suppose to have k a field such that $[k (\mu_n):k]=\phi (n)$ where $\mu_n$ is ...
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### What does $\mathbb{Q}(\sqrt{2},\sqrt{3})$ mean? [duplicate]

What set is $\mathbb{Q}(\sqrt{2},\sqrt{3})$? Is it the set $X = \{a\sqrt{2}+b\sqrt{3}:a,b\in\mathbb{Q}\}$?
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### When is the norm of a number even?

In the ring $$\textbf{Z}[i],$$ if the norm of an element is divisible by $2$, then the element must be divisible by $$1 + i,$$ and vice versa. A similar result holds for $$\textbf{Z}[\sqrt 3]$$ and ...
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### Number of solutions a polynomial can have as a function of the field?

Is there any limitation (upper bound) for number of solutions of polynomial equations? Having a background in engineering, my knowledge of higher algebra is rather limited, but I do know of ...
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### Number of Subfields sandwiched between two fields

Let $\omega$ be a complex cube root of unity such that $\omega \neq 1$. Suppose L is the field $\mathbb Q(2^{1/3},\omega)$ generated by them over the field of rationals. Then, the number of subfields ...
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### A reduction to the finite degree case

I am stuck trying to understand a proof in Asymptotic Differential Algebra and Model Theory of Transseries by L. van den Dries, J. van der Hoeven and M. Aschenbrenner. The result is the following: ...
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### If $\mathbb{F}$ is a finite field such that for every $a \in \mathbb{F}$ the equation $x^2=a$ has a solution in $\mathbb{F}$

Let $\mathbb{F}$ be a finite field such that for every $a \in \mathbb{F}$ the equation $x^2=a$ has a solution in $\mathbb{F}$. Then $\mathbb{F} \simeq \mathbb{F}_{2^n}$. I have tried this for the ...
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### If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ necessarily transcendental over $\mathbb{Q}$?

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ is necessarily transcendental over $\mathbb{Q}$ ? In Wiki I found answer is no but I can't cook up an counter example. ...
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### Each exponent of each term of an irreducible polynomial is divisible by p

I'm studying the field theory,in particular, the separable extension. My question is the followings. WTS : an irreducible polynomial q(x) over a field F of characteristic p≠0 is not separable iff ...
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### On every other finite field at least one of −1, 2 and −2 is a square, because the product of two non squares is a square

[Except on field extensions of $\mathbb{F}_2$] On every other finite field at least one of $−1$, $2$ and $−2$ is a square, because the product of two non squares is a square. I don't see why this is ...
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### Isomorphic fields of finite degree have same dimension over base field

Let $K/F$ be a field extension and $L_1,L_2$ subfields of $K$ such that $L_1$ and $L_2$ have finite degree over $F$. Does $L_1 \cong L_2$ imply $[L_1 : F ]=[L_2 : F]$? Obviously, if the isomorphism ...
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### Let $\omega= \cos \frac{2\pi}{10}+i\sin \frac{2\pi}{10}$, let $K=\mathbb{Q}(\omega^2)$ and $L=\mathbb{Q}(\omega)$. [closed]

Let $\omega= \cos \frac{2\pi}{10}+i\sin \frac{2\pi}{10}$. Let $K=\mathbb{Q}(\omega^2)$ and $L=\mathbb{Q}(\omega)$. Then A. $[L,\mathbb{Q}]=10$ B. $[L,K]=2$ C. $[K,\mathbb{Q}]=4$ D. $L=K$
### Proving $\mathbb R[x]/\langle 1+x^2\rangle$ $\cong$ $\mathbb C$ without using 1st isomorphism theorem
I've seen many the proofs of this by making use of First isomorphism theorem, by considering the map,$$\phi:\mathbb R[x]\rightarrow\mathbb C$$ defined by $\phi(a+bx)=a+bi$. My questions are ...