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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2answers
41 views

Showing that every polynomial over the Algebraic Numbers has a $0$ in the Algebraic Numbers. [duplicate]

Let $\mathbb{A}$ denote the field of Algebraic Numbers: the field of all complex numbers that are algebraic over $\mathbb{Q}$. Assuming that every polynomial over $\mathbb{C}$ has a $0$ in ...
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1answer
28 views

Cyclic extension of degree $2$ of $\mathbb{F_2}$

I want to find a cyclic extension of degree $2$ of $\mathbb{F_2}$ The degree of its minimal polynomial is $2$, i.e. a quadratic polynomial. The only cyclic field of degree $2$ which I can think of ...
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2answers
89 views

Geometric reasons finite fields have prime power orders?

All variations of proofs that finite fields have prime power orders have a very algebraic feel to them. I was wondering - is there a more geometric way to see why this is true?
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3answers
94 views

Can every function be represented as polynomial [closed]

Can every function $f: R^n \to R^n$ or$R^n \to R$ be represented as polynomial either of degree $n$ or infinite degree. Are there any proofs to this statement if it is true?If it has no Taylor-Power ...
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1answer
31 views

Normal closure $L$ of $\mathbb{Q}(\sqrt[4]{7})$ and structure of $Gal(L/\mathbb{Q})$

I think I have done (a) but I need some guidance on (b), if possible (a). Find a normal closure $L$ of $\mathbb{Q}(\sqrt[4]{7})$ To construct the normal closure I could adjoin the roots of ...
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0answers
130 views

Does the category of algebraically closed fields of characteristic $p$ change when $p$ changes?

EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here. Let $\mathrm{ACF}_p$ denote the category of algebraically ...
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2answers
54 views

$K= \mathbb{F}_2(\alpha)$ $\alpha$ root of $X^4+X+1 \in \mathbb{F}_2[X]$. Find degree and minimal polynomial

Question 1: Find $[K:\mathbb{F_2}]$ Idea: I have tried looking at the irreducibility of the polynomial, $X^4+X+1 $ and have so far been unsuccessful. Is there another way to do this apart from using ...
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1answer
19 views

Principal Ideals

Let $R$ be a commutative ring with unity. I'm trying to prove if every ideal of $R[X]$ is a principal ideal, then $R$ is a field. So it's sufficient to show $R$ is a division ring. Question: What ...
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0answers
22 views

is it true that closures preserve isomorphisms [duplicate]

Suppose I have two isomorphic integral domains $A$ and $B$. Are the fields of fractions of these two rings isomorphic as well?
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1answer
87 views

Prove that $\mathbb Q[\sqrt[3]2]$ is a field

We define the set: $$\mathbb{Q}[\sqrt[3]2]=\{a_{0}+a_{1}\sqrt[3]{2}+a_{2}\sqrt[3]{2^{2}}:a_{0}, a_1,a_2\in\mathbb{Q}\}$$ It's easy to prove all the properties of fields, except for the unit ...
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1answer
49 views

Galois group of the field extension

Determine the Galois group of the field extension $E/\mathbb{Q}$, where $E$ is the splitting field of the polynomial $x^4-2\in \mathbb{Q}[x] $. Here it is clear that $\Bbb Q(\sqrt[4]{2})$ is a ...
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1answer
23 views

Extention of a field

I am studying about diffrent extentions of a field like F[x], and I have a problem to undrestand how the quotient which is generated by ideal p(x), (p(x) is an irreducible polynomiyal in F[X]) extends ...
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1answer
47 views

Quadratic Extensions and Dihedral Galois Group

Let $f(x)$ be an irreducible polynomial of degree $4$ with rational coefficients, let $\alpha$ be a root of $f$ and set $L=\mathbb{Q}(\alpha)$ (say $\alpha \in \mathbb{C}$). Let $K$ be the splitting ...
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2answers
43 views

Confusion with Galois Group

The more I progress, the more contradictions or ambiguity I come by. Probably because Galois group builds up and numerous obscure and abstract concepts and being wobbly and one of them causes tragedy. ...
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0answers
63 views

What is the number of subgroups of $C_2 \times C_2 \times C_2 \times \cdots \times C_2$?

I think that counting the number of subgroups of various groups is usually very difficult. I was wondering about the number of subgroups of $(C_2)^n$. For example, there are 5 subgroups of $C_2 ...
2
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3answers
34 views

Extension of finite fields under norm map

If $L/K$ is a finite Galois extension with group $G$, we can define the norm of an element $a\in L$ as \begin{equation} N_{L/K}(a)=\prod_{\sigma\in G}\sigma(a). \end{equation} And obviously, this ...
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0answers
9 views

Elements of finite field extensions [duplicate]

I posted this question earlier, and was wondering if somebody could help me answer it: Existence of $f'\in F$ such that $f^{[G:F]}=(f')^n$ for finite extension $F\subset G$ Thanks in ...
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1answer
18 views

Hilbert's Nullstellensatz - question about generalization

It is well-known that if $k$ is algebraically closed field that has infinite transcendence degree over the prime field $\mathbb{Q}$ or $\mathbb{F}_p$ then the maximal ideals of $k[x_1,...,x_n]$ are of ...
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1answer
29 views

What is the smallest $m>0$ for the Frobenius automorhism of a Galois Field to be the identity?

Surprisingly scarce information on this particular problem. $\phi$ is the Frobenius automorphism of $GF(p^n)$ for some prime $p$. Find the smallest $m>0$ such that $\phi^m$ is the identity ...
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0answers
21 views

Finite Groups as Galois Groups of Field Extensions [duplicate]

Let $F$ be a field, and $x_{1}, \dots, x_{n}$ free variables. Let $K=F(x_{1},x_{2},\dots,x_{n})$, being the fraction field of the ring of polynomials in the variables $x_{1},\dots,x_{n}$. Show that ...
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1answer
23 views

Jordan-Chevalley decomposition for non-algebraically closed fields?

Say that we have a field $\mathbb{K}$ which is not necessarily algebraically closed, and $V$ a finite-dimensional vector space over $\mathbb{K}$. Given an endomorphism $X \in ...
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2answers
36 views

Field extension with odd degree

Let $F(a)$ be Field extension over $F$ such that $[F(a):F]=5$. I know that if $[F(a) : F]$ is odd then $F(a) = F(a^2)$. So how can I show that $F(a)= F(a^2+a+1)$. Could somebody please give me hints. ...
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2answers
34 views

Galois Extension with Galois Group $(\mathbb{Z}/2\mathbb{Z})^{3}$

Write an example of a Galois extension of fields that has as a Galois group $(\mathbb{Z}/2\mathbb{Z})^{3}$ I'm not very familiar with Galois theory, so I don't know of a general procedure to ...
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2answers
40 views

Rational Polynomial of Degree $3$ satisfying $2\cos{(2\pi/7)}$

Let $\eta = \zeta_{7}+\bar{\zeta_{7}}$, for $\zeta_{7}=\exp{(2i\pi/7)}$. Find a polynomial of degree 3 with rational coefficients that $\eta$ satisfies. I'm not so sure on how to begin by this ...
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2answers
50 views

What does the terminology “characteristic polynomial” mean?

In the documentation of the command "MininimalPolynomial" in Mathematica it says "Find the characteristic polynomial of Sqrt[2] over the extension E^(I Pi/4). The code: MinimalPolynomial[Sqrt[2], x, ...
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1answer
24 views

$a$ is algebraic over $k(b)$ where $b=g(a)$ for some non constant polynomial $g$ [closed]

Let $k \subset K$ be a field extension and $a \in K$. Show that if $g \in k[x]$ is any nonconstant polynomial and $b = g(a)$, then $a$ is algebraic over $k(b)$.
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2answers
43 views

Is $K[a,b]=K(a,b)$ for algebraic $a, b$?

Consider a field extension $L/K$ and $a,b \in L$ algebraic over K. I know that $K[a]=K(a)$ if and only if $a$ is algebraic over $K$. Is it true that in this setting $K[a,b]$ is already a field?
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1answer
41 views

A polynomial with a root in $\mathbb{F}_p \ \forall p$, where $p$ is prime, but no root in $\mathbb{Z}$ [duplicate]

Give an example of a polynomial $f(x) \in \mathbb{Z}[x]$ which has a root in every finite field $\mathbb{F}_p$, but no root in $\mathbb{Z}$.
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3answers
47 views

Show that $\mathbb{Q}(\sqrt2, \sqrt[3]2)$ is a primitive field extension of $\mathbb{Q}$.

I've tried a method similar to showing that $\mathbb{Q}(\sqrt2, \sqrt3)$ is a primitive field extension, but the cube root of 2 just makes it a nightmare. Thanks in advance
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1answer
30 views

Intersection of field extensions

Let $F$ be a field and $K$ a field extension of $F$. Suppose $a,b\in K$ are algebraic over $F$ with degrees $m$ and $n$, where $m,n$ are relatively prime. Then $F(a) \cap F(b) = F$. I see that the ...
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1answer
22 views

Quotient of polynomial ring over an algebraically closed field, isomorphism of fields

Let $K$ be an algebraically closed field. Let $n$ be an integer. Let $M$ be a maximal ideal of the ring of polynomial $K[X_1,...,X_n]$. Then the quotient ring $K[X_1,...,X_n]/M$ a field and a ...
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2answers
46 views

Why does $F(\sqrt{a+b+2\sqrt{ab}}) = F(\sqrt{a},\sqrt{b})$?

Let $F$ be a field of characteristic $\neq 2$. Let $a \neq b \in F$, and $F(\sqrt{a},\sqrt{b})$ is of degree 4 over $F$. I've shown that $F(\sqrt{a}+\sqrt{b}) = F(\sqrt{a},\sqrt{b})$. Observe that ...
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2answers
23 views

One basis of extension field works for all roots of minimum polynomial?

Let $p(x)$ be a minimum polynomial of $c \in K \supset F$ over $F[x]$. Say $p(x)$ has multiple distinct roots $\{r_i\}$including $c$. I know that $F(c) \cong F(r_i)$ for each $i$, but is a basis of ...
0
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1answer
62 views

Existence of $f'\in F$ such that $f^{[G:F]}=(f')^n$ for finite extension $F\subset G$

Let $F$ be a field and $u\in F$. Further, let $F\subset G$ be a finite extension and $v\in G$ such that $v^n=u$ for $n>0$. How would I show that there exists a $\tilde{u}\in F$ such that ...
1
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1answer
45 views

Extension of $\mathbb{Q}$ over which $\pi$ is algebraic of degree 3?

I think $\pi$ is algebraic of degree 3 over $\mathbb{Q}(\pi^3)$. To prove it, I need to show that $\pi \notin \mathbb{Q}(\pi^3)$ (which implies that $x^3-\pi^3$ is the minimum polynomial of $\pi$ over ...
1
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3answers
89 views

Finding basis of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ over $\mathbb{Q}$

I'm having trouble finding a basis for $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$. So far I know that $[K=\mathbb{Q}(\sqrt{2}):\mathbb{Q}] = 2$ and $[L=\mathbb{Q}(\sqrt{2}, \sqrt{3}):K] = 2$, but it's ...
1
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1answer
39 views

Show $x^2-2$ is irreducible over $\mathbb{Q}(\sqrt[3]{4})$

I'm having trouble showing that $p(x)=x^2-2$ is irreducible over $\mathbb{Q}(\sqrt[3]{4})$. What I tried: I already know the roots of $p(x)$ are $r=\pm \sqrt{2}$, so I just need to show that ...
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0answers
16 views

$ \tan$ function of two arguments when second part is modified

Can $ \tan( x + n \,y)$ be expressed as a function of $ \tan x, \tan y $ and some $ f(n) ? $ A particular integer validity for $n$ in: $$\tan( x + y \cos n \pi) = \dfrac{\tan x + \ f(n) \tan y ...
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3answers
61 views

How is it true that $-\frac{1}{2}+i\frac{\sqrt{3}}{2} \in \mathbb{Q}(i,\sqrt{3})$?

I am having doubts this is true. So elements of $\mathbb{Q}(i,\sqrt{3})$has the form $p+qi+r\sqrt{3}$ for some rational $p,q,r$. This means that I must be able to express ...
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1answer
32 views

If $R$ is an integral domain, show that there is no subfield $k$ of Frac($R$) containing $R$.

If $R$ is an integral domain, show that there is no subfield $k$ of Frac($R$) containing $R$. I think the way to prove this is by contradiction. So, let $R$ be an integral domain, and let $k$ be a ...
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1answer
61 views

Pythagorean Closure of $\mathbb{Q}$ is extremely obscure. Can someone give a clear answer?

My previous question didn't get me a definite answer. I've refined it so with a specific example so it might help someone clear my confusion. In Stewart's book, The Pythagorean closure is defined as ...
1
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1answer
57 views

Show that $\sqrt[15]{63}$ does not belong in $\mathbb Q(\sqrt[189]{147})$ [closed]

How to prove that $\sqrt[15]{63}$ does not belong in $\mathbb Q(\sqrt[189]{147})$?
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0answers
47 views

Showing that the field of rational functions is not dense

I am going through Counter Examples of Analysis but I am having trouble understanding a claim it makes. The book establishes that the set of rational functions defines an ordered field where the ...
1
vote
1answer
30 views

Irreducibility over field extension $\mathbb{Q}(\sqrt[3]{2})$

I want to show that $p(x)=x^2-1-\sqrt[3]{2}$ is irreducible over $\mathbb{Q}(\sqrt[3]{2})$. What I tried: Assume by contradiction that $p(x)$ is reducible; then it has a root $r$. We know that $\{1, ...
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4answers
108 views

Subgroup generated by $1 - \sqrt{2}$, $2 - \sqrt{3}$, $\sqrt{3} - \sqrt{2}$

For a number field $K$, Dirichlet's unit theorem says that$$(O_K)^\times = \mathbb{Z}^{r - 1} \oplus (\text{a finite cyclic group}),$$where $r$ is the number of all infinite places of $K$. An infinite ...
2
votes
1answer
34 views

What is $\left(\overline{\mathbb{C}(t)}\right)^{\times}$?

In a previous question, I asked about the automorphism groups of the groups of units of various fields. In an answer to this question it is explained that $\mathbb{C}^{\times} \cong S^{1} \oplus ...
2
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6answers
103 views

Why is $\mathbb{Q}[\sqrt{2}]$ a field?

It seems to me that the hard part of this proof is multiplicative inverses. I know how to prove this by considering the multiplication of two arbitrary elements: $$(a + b\sqrt{2})(a' + b'\sqrt{2}) = ...
0
votes
2answers
26 views

How to prove $\alpha =\{1, x,…,x^n\}$ is a basis for $P_n(F)$

Let $v=a_0+a_1x+...+a_nx^n$, if $a_0+a_1x+...+a_nx^n=0$, then all coefficient $a_i\in F$ are $0$ as variables are all with different degrees so it is linearly independent (Here I am no sure how to ...
0
votes
1answer
29 views

How to verify that $P_n(F)$ is a vector space over field $F$?

I assume $p(x)=a_nx^n+...+a_0$ where $a_i\in F$ and $x$ is variable. To verify it as vector space, I think we need to check axioms. i.e. $\forall x, y,z\in V,\ (x+y)+z=x+(y+z)$ and $\forall x, y\in ...
2
votes
2answers
52 views

Field extensions of $\mathbb{Q}$, $\mathbb{Q}(\xi_7)$ and $\mathbb{Q}(\xi_7+\xi_7^{-1})$

Let $\xi_7$ denote the complex number $e^{2\pi i/7}$ and let $\beta = \xi_7+\xi_7^{-1}$, consider the field extensions $\mathbb{Q} \subset \mathbb{Q}(\beta) \subset \mathbb{Q}(\xi_7) $. Determine ...