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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1answer
54 views

is f(x) = 0 irreducible in $\mathbb{Z} /2 \mathbb{Z}$?

Let's say I have a polynomial like $$f(x) = 4x^2 +12x +28$$ when I reduce this with respect to mod 2; I end up with $0$. Can I say that zero is irreducible in ...
0
votes
1answer
19 views

For $f\in\mathbb{Q}[x]$, Gal($f)\subset S_n$ is a subset of $A_n$ iff $\Delta(f)$ is a square in $\mathbb{Q}^*$

Let $f\in \mathbb{Q}[x]$ a monic irreducible polynomial, and Gal($f$) be a subgroup of $S_n$. How do I prove that Gal($f$) $\subset A_n\iff \Delta(f)$ is a square in $\mathbb{Q}^*$? I know what ...
2
votes
1answer
33 views

intermediate Field extension by irreducible polynomial

Thanks for any help or comments. Suppose $f(x)\in F[x]$ is an irreducible polynomial of composite degree $n$. So there exist an closure field $\bar{F}$ such that $f(x)$ is completely reducible in ...
2
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1answer
34 views

On the action of galois groups in towers of fields

I would like some confirmation on certain statements I believe to be true: Let $K\subset L\subset M$ be a tower of fields such that both extensions $L/K$ and $M/K$ are galois. Let $f(x) \in K[x]$ be ...
2
votes
1answer
41 views

Show that the degree of any irreducible factor of $x^8-x$ over $\mathbb Z_2$ is $1$ or $3$

Statement Prove that the degree of any irreducible factor of $x^8-x$ over $\mathbb Z_2$ is $1$ or $3$. The hint in the back of the book makes a certain claim that I wasn't sure about. The author ...
6
votes
0answers
79 views

If $E/F$ is finite, $F$ has no odd extensions, and $E$ has no extensions of degree $2$, then $F$ is perfect and $E$ is algebraically closed?

I'm struggling to understand the claim that if $E/F$ is a finite field extension, $F$ has no odd extensions, and $E$ has no extensions of degree $2$, then $F$ is perfect and $E$ is algebraically ...
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2answers
46 views

Galois group of the splitting field of the minimal polynomial over $\Bbb{Q}$

Determine the Galois group of the splitting field of the minimal polynomial of the following algebraic numbers $\sqrt{2}+\sqrt{3}+\sqrt{6}$ over $\Bbb{Q}$. It is clear that ...
6
votes
3answers
281 views

A field has only one isomorphic subfield to itself?

Let $E$ be a field and $F$ be a subfield of $E$ which is isomorphic to $E$. Then is $F$ equal to $E$? It seems to be clear but I couldn't prove it. Could you please explain this statement?
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1answer
42 views

Complex numbers of degree less than or equal to 2 over rational numbers

Im trying to figure out which complex numbers have degree $\leq$2 over $\mathbb{Q}$ and then figure out which have degree $\leq$2 over $\mathbb{R}$. For the first question, I know that it is at least ...
0
votes
0answers
27 views

Showing fields are algebraically closed

Let K be a field, and let P be separable and irreducible over K. Let L be a splitting field of P over K. I want to show that the fields K(u) and K(v) are isomorphic, where u and v are roots of P, ...
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1answer
41 views

Prove that $\mathbb{F}_p(x,y)/\mathbb{F}_p(x^p,y^p)$ is not a simple extension [closed]

Prove that $\mathbb{F}_p(x,y)/ \mathbb{F}_p(x^p,y^p)$ is not a simple extension by explicitly exhibiting an infinite number of intermediate subfields .
1
vote
1answer
33 views

Roots of polynomial in field extension

My question is stated below: Let $K=\mathbb Z_3 [x]$ and $p(x) ∈ \mathbb Z_3 [x]$ be defined by $p(x) = x^4 + x + 2$. Consider the field extension $\mathbb Z_3 [x]/(p(x))$. Define $q(x) ∈ \mathbb ...
22
votes
7answers
2k views

How to prove that a complex number is not a root of unity?

$\frac35+i\frac45$ is not a root of unity though its absolute value is $1$. Suppose I don't have a calculator to calculate out its argument then how do I prove it? Is there any approach from ...
2
votes
0answers
22 views

Transcendental extension not isomorphic to its closure

Suppose I'm given a field extension $K/F$ with $\alpha\in K$ transcendental over $F$, the claim is that $F(x)\cong F(\alpha)$. It's a statement without proof in our class notes, and the remarks ...
0
votes
1answer
21 views

Potentially diagonal $K$-algebra

In our abstract algebra class, we had two lectures on field extensions. Then we were given the following homework problem (with this problem on diagonal $K$-algebra). Let $K$ be a field. A ...
3
votes
2answers
49 views

Find the minimal polynomial of $\sqrt{3} +i$ over $Q(i)$ and over $Q(\sqrt{3})$

Find the minimum polynomial of $\sqrt{3} +i$ over $Q(i)$ and $Q(\sqrt{3})$. My solution: Over $Q(i)$ : Suppose $a = \sqrt{3} +i$, $a-\sqrt{3} =i$; so, the minimal polynomial is $x-\sqrt{3} ...
1
vote
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 ...
0
votes
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 ...
2
votes
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?
1
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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 ...
0
votes
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 ...
12
votes
0answers
131 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 ...
3
votes
2answers
57 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 ...
1
vote
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 ...
0
votes
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?
1
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1answer
90 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 ...
3
votes
1answer
51 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 ...
0
votes
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 ...
1
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1answer
50 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 ...
1
vote
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. ...
2
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
2answers
37 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. ...
0
votes
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 ...
2
votes
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 ...
0
votes
2answers
51 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, ...
0
votes
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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votes
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?
2
votes
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
48 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
3
votes
1answer
31 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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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
votes
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 ...