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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Exercise 5 .39 page 46 Real and Abstract Analysis [Hewitt and Stromberg]

How to show that every non-Archimedian field contains a subfield algebraically and order isomorphic to $\mathbb{Q}(t)$, where $\mathbb{Q}(t)$ is the field of rational functions with coefficient in ...
6
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1answer
75 views

Is the extension Galois if $\mathrm{Aut}(K)$ acts transitively on the non-ramified prime ideals?

Let $K/\mathbb Q$ be a finite extension such that $\mathrm{Aut}(K)$ acts transitively on the prime ideals that are not ramified above the same prime $p\in\mathbb N$. Is $K$ Galois? Thanks in advance. ...
2
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0answers
66 views

Fields whose algebraic closure cannot be constructed without the axiom of choice

One can show that the statement that every field has an algebraic closure requires the axiom of choice. However, for almost all "everyday" fields, it seems that one can actually produce an algebraic ...
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1answer
32 views

Field Extensions, Subfield

Let $F$ be a subfield of a field $K$ and let $t \in K$ Let $t$ be algebraic of degree $n>1$ over $F$. Prove that $[K:F] \ge n$ Clearly there exists a polynomial $P(x)$ such that ...
1
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3answers
66 views

Show that a map is not an automorphism in an infinite field

How should I show that a map $f(x) = x^{-1}$ for $x \neq 0$ and $f(0) = 0$ is not an automorphism for an infinite field? Thanks for any hints. Kuba
1
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1answer
38 views

The characteristic of the field $GF(p^n)$

How to show that characteristic of the field $GF(p^n)$ is $p$? I have come across this fact on Wikipedia webpage, but don't know how to prove it. Thanks
2
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1answer
32 views

complex automorphisms acting on a projective variety

Consider a complex projective variety $X=\operatorname{Proj}\frac{\mathbb C[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}$ with $f_1,\ldots,f_n$ homogeneous polynomials. If $\sigma\in\operatorname{Aut}(\mathbb ...
2
votes
1answer
39 views

Polynomials in $Z_p[x]/f(x)$

For shorthand, suppose $K=\mathbb{Z}_p[x]/f(x)$, $p$ a prime, and $\deg(f)=n$ where $f\in \mathbb{Z}_p[x]$. Then, how do we show that (1) $K$ can be written as $\mathbb{Z}_p[\theta]$, where $\theta$ ...
1
vote
1answer
83 views

Existence of an automorphism of $\Bbb C$ that fixes a finite set pointwise but does not fix $\Bbb R$ setwise

I would like to construct a (ring-theoretic) automorphism of $\Bbb C$ that fixes a finite set $A$ pointwise but does not fix $\Bbb R$ setwise. Marker's Model Theory, Corollary 1.3.6 does that in this ...
4
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1answer
58 views

Subgroups of $F^*$ are cyclic

Q: If $F$ a field then every finite subgroup of $F^*$ is cyclic. Solution: Suppose $d\mid |G|$ for $G$ subgroup of $F^*$ and $G$ not cyclic. Suppose $A,B$ subgroups of $G$ of order $d$. Then ...
0
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1answer
14 views

There are finite distinct restrictions to a subfield

Consider the field extension $L\subseteq K\subseteq \mathbb C$ where $K/L$ is finite. I must show that the set $\{\sigma_{|K}\,:\,\sigma\in\operatorname{Gal}( \mathbb C/L)\}$ is finite, but I have ...
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0answers
19 views

Discriminant of a splitting field [duplicate]

Let $f(x)\in\mathbf Q[x]$ and let $K$ be the splitting field of $f(x)$. How is the discriminant of $K$ (as a number field) related to the discriminant of $f(x)$? Are they divisible by the same primes? ...
0
votes
1answer
30 views

Finitely generated extension of a countable field

Suppose that $K$ is a countable field, and consider some elements $a_1,\ldots,a_n$ that lie in some extension of $K$. Can I conclude that $K(a_1,\ldots,a_n)$ is countable field? Thanks in advance
1
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0answers
33 views

simple extension with algebraic over the field

Assume that $F$ is infinite, that $v,w \in K$ are algebraic over $F$, and that $w$ is a root of a separable polynomial in $F[x]$. How will I be able to prove that $F(v,w)$ is a simple extension of ...
3
votes
2answers
62 views

Minimal polynomial of $\sqrt{2}+\sqrt[3]{3}$.

Is there a way to show that the minimal polynomial of this number over $\mathbb Q$ has degree $6$ without too much annoying computations? I have "reduced" it to showing that both $\sqrt{2}$ and ...
2
votes
3answers
46 views

How to deduce whether $\sqrt{3}i \in \mathbb Q (u)$ or not, where u is a root of $x^3-2$

How can one deduce whether $\sqrt{3}i \in \mathbb Q (u)$ or not, where u is a root of $x^3-2$ ? $\mathbb Q (u) = \{c_0 +c_1u+c_2u^2 | c_i \in \mathbb Q\}$ Is it a trivial computation, or does this ...
0
votes
0answers
71 views

Prove that if $a^2 + ab + b^2 = 0$ then $a = b = 0$?

We are given that $a, b \in F_{2^n}$ where $n$ is an odd +ve integer. Suppose $a^2 + ab + b^2 = 0$ then we have either $a = 2^n-b^2$ or $a+b = 2^n - b^2$. Which implies that $\sqrt{2^n -a} = +-b $ or ...
1
vote
1answer
27 views

Automorphism group of a non-normal field extension

Consider finite field extensions $L>K>F$ such that $L/F$ is Galois, and $K/F$ is separable. I am particularly interested in the case $F=\mathbb{Q}$. By Galois theory, $K/F$ is normal iff ...
1
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3answers
68 views

Prove that every extension of a finite field is normal

In book by Roman 'Field Theory' it is written that it is straightforward that every extension of a finite field is normal. However I just cannot see it. Can you help me with this problem? Thank
8
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2answers
102 views

Order of invertible matrices

I recently came across an interesting problem in Artin which says: If $A \in GL_2(\mathbb{Z})$ is of finite order then it has order $1,2,3,4,6$. I was looking for a generalization of this problem. ...
5
votes
1answer
70 views

$Gal(\mathbb{Q}(\sqrt 2 + \sqrt 3)/\mathbb{Q})$

A basis for $\mathbb{Q}(\sqrt 2 + \sqrt 3)$ over $\mathbb{Q}$ is $\{1,\sqrt 2 , \sqrt 3 , \sqrt 6 \}$ The roots of $x^2 -2$ are $\pm \sqrt 2$ and the roots of $x^2 -3$ are $\pm \sqrt 3$ so to find ...
0
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1answer
52 views

finite field extension and normality

If $K$ is an extension field of $F$ such that $[K:F]=2$. Then $K$ is normal? I know that if $[K:F]=2$ then $K=F(u)$ where $u$ is the root of $f \in F[x]$. But how do you prove that dimension $2$ ...
4
votes
1answer
42 views

Why is the order of a prime element well-defined?

This is in relation to the $p$-adic valuation on the field of fractions $F$ of an integral domain $D$. The idea is that for each $x \in F$ there is a unique maximal $k$ such that $x = p^k u v^{-1}$ ...
1
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1answer
34 views

Rupture field of $X^p+T$ equals its splitting field [closed]

Let $K$ be a field of prime characteristic $p$. Let $P(X)=X^p+T$ be a polynomial from $K(T)[X]$. $P$ is irreducible over $K(T)$ by Eisenstein criterion. Show that a rupture field of $P$ is also a ...
0
votes
2answers
74 views

Field Extension Question: $K(a,b) \ne K(a+b)$

I am trying to come up with an example of when an extension $K(a,b)$ with $a,b$ in $E$ is not equal to $K(a+b)$. In short, $$K(a,b) \ne K(a+b).$$ I am learning abstract algebra for the first time, ...
1
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0answers
29 views

Reference for Galois Theory of infinite field extensions.

I would like to ask what is in your opinion the best place to learn about Infinite Galois Theory that requires not much knowledge of topology. I am searching for a text that explains the notions ...
4
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0answers
67 views

Isomorphism between finite fields adjoining a root

Let $p(x)=x^3+x^2+1$ and $q(x)=x^3+x+1$ be polynomials over the field $\mathbb{Z}_2$. Let $\alpha$ be a root of $p(x)$ and $\beta$ be a root of $q(x)$. Now let $K=\mathbb{Z}_2(\alpha)$ and ...
0
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0answers
15 views

Consider $n$ a product of distinct primes $p_j$, with each $p_j-1$ a power of 2, then the regular $n$-gon is constructible.

I am having trouble showing that if $n$ is a product of distinct primes $p_j$, with each $p_j-1$ a power of $2$, then the regular $n$-gon is constructible. Apparently it could be useful to use that ...
2
votes
1answer
43 views

Embedding a finite extension of $F(X)$ into a pure transcendental extension

If $F$ is any algebraically closed field, and $L \supset F(X)$ is a finite extension of the purely transcendental extension of $F$ of transcendence degree $1$, then can $L$ necessarily be embedded ...
2
votes
1answer
62 views

Structure of a module and finite fields

I am trying the following problem which appeared on an exam for a grad course: Determine the structure of finite field $F$ as a module over $F_p[x]$ when $xv=Lv$. Here $L(z)=z^p$ and ...
0
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1answer
17 views

a question about normal extensions

Let $E$ be a finite normal extension of the field $F$, for which the Galois group $G(E/F)$ is Abelian, and let $K$ be a field intermediate to $F$ and $E$. Let $a\in K$ and $f(x)\in F[x]$ be the ...
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1answer
79 views

What is zero times zero

What is zero zeros? What are no nothings? From a mathematical point of view it would be my thing, but none of us are educated that much in math, so I am curious to hear an expert opinion.
1
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1answer
34 views

Prove that $B$ is a subfield of $F$

If a subring $B$ of a field $F$ is closed with respect to multiplicative inverses, then $B$ is a field. Fields are commutative rings with unity, and every nonzero element has an inverse. A subring ...
2
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0answers
31 views

Number of elements and number of different basis of $\mathbb F_5^3$

Let $\mathbb F:=\mathbb F_5$ the field with five elements. (i) How many elements has $\mathbb F^3$? (ii) How many different basis has $\mathbb F^3$? My idea: (i) $\mathbb F^3$ has $5^3$ elements. ...
0
votes
1answer
21 views

Basis for field extension by an algebraic element

Is was wondering if, given a field F with a known basis and an element b which is algebraic over that field, it is possible to construct explicitly a basis for F[b], the extension of F by b. Suppose, ...
2
votes
1answer
79 views

Why isn't $e^{\frac{2\pi i}{9}}$ an element of $\mathbb{Q}(\sqrt[9]{2}, e^{\frac{2\pi i}{3}})$?

The question is in the headline above. I need to know this, because I want to show that $\mathbb{Q}(\sqrt[9]{2}, e^{\frac{2\pi i}{3}})/\mathbb{Q}$ is not a normal extension and it's quite obvious I ...
1
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1answer
54 views

Choosing a polynomial for CRC

CRC checksum is a homomorphism from polynomials over $\mathbb F_2$ to itself. As I understand, the map $f\mapsto g$ it is simply remainder from division $f$ by $p$, where $p$ is a fixed polynomial for ...
2
votes
2answers
40 views

Polynomial ring and extension field

Let $K$ be a field and $p(x) \in K[x]$ a monic irreducible polynomial of degree $n$, suppose $F/K$ is a field extension, and there exist $u \in K[x]$ which is a root of $p(x)$. 1) Let $K(u)$ be the ...
0
votes
2answers
70 views

Calculate the degree of a field extension

Find degree $[\mathbb{Q}(i,\sqrt{2}) : \mathbb{Q}]$ let $a = \sqrt{2}$ $a^2 = 2$ $\therefore a$ is a root of $q(x) = x^2 -2$, where $q(x)\in\mathbb{Q}(i\sqrt{2})[x]$ means degree of $a$ over ...
0
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1answer
32 views

Proving that operations give equal results given equal inputs

I was reading about the 9 or 12 basic properties of 'fields' (if that's what they're called) in a book called Spivak's Calculus, 3rd Edition, and got quite befuddled by dealing with as basic stuff as ...
0
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1answer
37 views

Show that two field extensions are the same

Can you help me with showing that these two field extensions are the same: $\mathbb{Q}(\sqrt{3}, \sqrt[3]{5})$ $\mathbb{Q}(\sqrt{3} + b\sqrt[3]{5})$, where $b\neq0$ is any rational number. Thanks ...
2
votes
1answer
17 views

Computing determinants using derivatives in an arbitrary field

When computing determinants that depend on a parameter $t\in \Bbb R$, it is often useful to use the fact that \begin{align} \det(V_1(t),\dots,V_n(t))&=\det(V_1(a),\dots,V_n(a))+\\ ...
3
votes
3answers
127 views

Check that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational

How to prove that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational. I will appreciate any proof, but I had such exercise during lecture in field theory. Thanks.
1
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1answer
49 views

Find a subfield of $\mathbb{C}$ isomorphic to other field

Do you know, how I can find a subfield of $\mathbb{C}$ isomorphic to $F = \mathbb{Q}(\sqrt[3]{7})$ such that $F \nsubseteq \mathbb{R}$? I don't even have clue, how I should start. Thanks
0
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1answer
38 views

nontrivial $K$-automorphism of $K(x)$

How can I find $K$-automorphism $\sigma \in Aut(K(x))$ different from identity such that $\sigma (x(x+1))=x(x+1)$?
3
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1answer
54 views

Is there a nice topology on Aut(C)?

Let $G=\mathrm{Aut}(\mathbf C)$, the group of field automorphisms of the complex numbers. It is a very large group (see this MSE question and the nice answer by Andres). For instance, there even ...
3
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0answers
20 views

A notion of transcendence degree for fields of formal power series?

There is an intuitive sense in which one would like to say that the field of formal Laurent series $k((z))$ over a field $k$ has "transcendence degree $1$". Of course, it doesn't have transcendence ...
1
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1answer
30 views

A doubt regarding splitting field.

$\Bbb Q(\omega)= \Bbb Q(\sqrt3,\iota)$ This is written in my text book that i am following. But I think this is a typo. Since $\Bbb Q(\sqrt3,\iota)$ is a larger field. in which $\Bbb Q(\omega)$ is ...
12
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2answers
517 views

Is number rational?

How can we check if number $a=\frac{ \sqrt[4]{2}+\sqrt[3]{3}}{\sqrt[4]{2}+\sqrt[3]{3} +1}$ is rational? Is there any smart solution? Another assignment is to find $\left( ...
0
votes
0answers
22 views

Karatsuba Method

For polynomials $f(x)$, $g(x)$ of degree $d = 2^{r-1}-1$, how do I check that multiplying $f(x)$ and $g(x)$ by the Karatsuba method requires $3^{r-1}$ multiplications in the field $F$?