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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83 views

How to prove $\mathbb{N}\subseteq\mathbb{R}$ given basic field axiomatisation.

I understand that $\mathbb{R}$ can be defined axiomatically in a number of different ways, including as an extension from more basic number systems (e.g., $\mathbb{N}$, $\mathbb{Z}$, or $\mathbb{Q}$) ...
2
votes
1answer
57 views

Commutativity of “extension” and “taking the radical” of ideals

Let $K$ be a field (not necessarily algebraically closed) and $\overline{K}$ its algebraic closure. By $K[\text{X}]$, I mean $K[X_1,...,X_n]$. Is it true that the operations of "extension" and ...
3
votes
1answer
34 views

Splitting field of $x^4-4x^3+4x^2-3$

I've got that $x^4-4x^3+4x^2-3 =(x^2-2x+ \sqrt{3})(x^2-2x-\sqrt{3})$ The roots of the polynomials are: $\alpha = 1+\sqrt{1-\sqrt{3}}$ $\quad$ $\alpha_1= 1-\sqrt{1-\sqrt{3}}$ $\quad$ $\beta= ...
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1answer
33 views

Expanding an expression in a certain field

If $\mathbb F_2$ is a field of characteristic $2$, then we have $x+x=y+y=z+z=0$ for all $x,y,z \in \mathbb F_2$. When I expand $(x+y)(y+z)(z+x)$, I get \begin{align} (x+y)(y+z)(z+x) &= ...
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1answer
24 views

Subring of a field [closed]

Let $R$ be a subring of a field $F$ such that for each $x\in F$ either $x\in R$ or $x^{-1}\in R$. Prove that if $I$ and $J$ are two ideals of $R$, then either $I\subseteq J$ or $J\subseteq I$.
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0answers
17 views

Completion of surreal subfields

Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $ < \kappa$. In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and ...
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1answer
59 views

Can you construct a field of characteristic $\neq 0, 2$ such that every one of its subrings is also a field?

A friend asked me this a few days ago, and I was thinking that it may be impossible, but now I'm not so sure. He suggested a "nonprincipal ultrapower" $(\mathbb{Z}/(2))^{N}$ such that every subring is ...
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0answers
19 views

Degree of irreducible polynomial

Let $\mathbb{F}$ be algebraic closure of field $\mathbb{k}$ and $[\mathbb{F}:\mathbb{k}] = n < \infty$. I have proved that if $f$ is irreducible over $\mathbb{k}$ than $\deg f | n$. But if $d | n$ ...
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0answers
57 views

I can't understand the formal definition of $\mathbb{R}$

I've always intuitively understood this set in intuitive sense, as "all numbers on the number line". However, now I want to know the formal definition: Consider the set of rational numbers, ...
1
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0answers
32 views

The sum of all primitive $n$-th roots of unity in the algebraic closure of $F(x)$

If $n$ is any natural number, then $\mu(n)$ is the sum of all primitive $n$-th roots of unity in $\mathbb{C}$ where $\mu$ is the Möbius function defined as $\mu(n)=0$ if $n$ is not square free and ...
2
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2answers
28 views

Finite extensions of $\mathbb F_p(t)$

Let $\mathbb F_p(t)$ the field of rational functions with coefficients in $\mathbb F_p$. Is it true or not that every finite extension $K$ of $\mathbb F_p(t)$ is $K\cong\mathbb F_{p^m}(t)$ for some ...
0
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1answer
28 views

Examples of fields in $\Bbb{R}$

I'm having trouble understanding how to determine fields I understand there are axioms it must satisfy to be considered a field like associativity, commutativity, distributivity, identity, and ...
0
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1answer
45 views

Why does multiplication turn $\mathbb{R}^2$ into $\mathbb{C}$?

Question: Show that multiplication makes $\mathbb{R}^2$ into a field (the field $\mathbb{C}$ of complex numbers) I know from another forum (Is $\mathbb R^2$ a field?) that $\mathbb{R}^2$ can be made ...
1
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1answer
21 views

Cardinality of collection of subfields of $\mathbb C$

The question is just curiosity on my part. The title says it all. I can see that the cardinality is at least $\aleph_1$ (take simple extensions by an uncountable family of transcendental numbers). But ...
0
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0answers
14 views

Bounds on the heights of the minimal polynomials of the algebraic coefficients of linear recurrence relations

Given a linear recurrence relation $$ a_n=c_1a_{n-1}+c_2a_{n-2}+\cdots+c_ka_{n-k} $$ with characteristic polynomial $$ ...
2
votes
2answers
51 views

Determine the degree of the extension $\mathbb{Q}(\sqrt{3 + 2\sqrt{2}})$.

The following is from a set of exercises and solutions. Determine the degree of the extension $\mathbb{Q}(\sqrt{3 + 2\sqrt{2}})$ over $\mathbb Q$. The solution says that the degree is $2$ since ...
2
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0answers
68 views

How to find galois group? E.g. $\mathrm{Gal}(\mathbb Q(\sqrt 2, \sqrt 3 , \sqrt 5 )/\mathbb Q$

I want to know how to find a Galois group. I know I need to look at automorphisms but I am not sure of the method. Could someone give me an idea, e.g. with the example: $\mathrm{Gal}(\mathbb Q(\sqrt ...
4
votes
1answer
32 views

Inseparable extensions

Given that $F/K$ is a finite extension of fields and the extension is not separable. My question is whether we can always find a subfield $L$ such that $K\subset L \subset F$ and $[F:L]=p$, where ...
5
votes
1answer
40 views

Showing that $f(x)^p=f(x^p)$ in field of characteristic $p$

I am trying to show that for any $f(x)\in F[x]$, where $F$ is a field of characteristic $p$, we have $f(x)^p=f(x^p)$. I figured that if $f(x)=\sum a_ix^i$, then $f(x)^p=\sum a_i^px^{ip}$ and ...
0
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1answer
17 views

How can we find $[GF(p^n):GF(p)]=n$?

I was searching why $[GF(p^n):GF(p)]=n$. It is not very logical, isn't it ? I know that $$GF(p^n)=\{x\in GF(p)^{alg}\mid x^{p^n}=x \}$$ is a field with $p^n$ element since it split $X^{p^n}-X$ which ...
0
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1answer
7 views

If $E$ is the splitting field of a separable polynomial on $K$, then $E/K$ is normale.

Let $K$ a field and $E$ the splitting field of a separable polynomial $f\in K[X]$. Show that $E/F$ is normale. My definition of normale extension is that $E/K$ is normale if ...
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1answer
34 views

Is $(\mathbb{Z}_{n},+_{n},._{n})$ a field, $\forall n\in \mathbb{N}$?

Is $(\mathbb{Z},+_{n},._{n})$ a field, $\forall n\in \mathbb{N}$? My answer is No, because for $n=6$, $(\mathbb{Z}_{6},+_{6},._{6})$ has a zero divisor but a field has no zero divisors so it can't be ...
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0answers
27 views

Minimal cyclotomic field containing a given quadratic field?

There was an exercise labeled difficile (English: difficult) in the material without solution: Suppose $d\in\mathbb Z\backslash\{0,1\}$ without square factors, and $n$ is the smallest natural ...
2
votes
3answers
50 views

Is $\mathbb{Z}[x]/\langle x-3\rangle$ a field?

Is $\mathbb{Z}[x]/\langle x-3\rangle$ a field? My thought: This quotient ring means that all polynomials in $\mathbb{Z}[x]$ are evaluated at $x=3$. So this is basically isomorphic to $\mathbb{Z}_3$ ...
0
votes
1answer
19 views

Let $F$ a field and $F^{alg}$ it algebraic closure. If $E/F$ is algebraic, does $E^{alg}=F^{alg}$ or not?

Let $F$ a field and $F^{alg}$ it algebraic closure. If $E/F$ is algebraic, does $E^{alg}=F^{alg}$ or not ? I would say yes, since every polynomial on $E$ split over $K^{alg}$ (since $E\subset ...
0
votes
1answer
32 views

Why the frobenius $\mathbb F_p^{alg}\longrightarrow \mathbb F_p^{alg}$ s.t. $x\longmapsto x^p$ is surjective?

Consider the frobenius $\mathbb F_p^{alg}\longrightarrow \mathbb F_p^{alg}$ defined by $x\longmapsto x^p$. 1) Why is it surjective ? I recall that $\mathbb F_p^{alg}$ is an algebraic closure of ...
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1answer
41 views

Why does an algebraically closed field not have any non-trivial algebraic field extensions?

Let $K$ be an algebraically closed field. Then there are no non-trivial algebraic field extensions of $K$. I can understand that if the field extension is of the form $K[x]/\langle p(x)\rangle$, ...
1
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2answers
34 views

A function in the integers module $p$ is polynomial. [closed]

Let $p$ a prime number and $\mathbb{F}_p$ the field of integers module $p$. Show that if $f:\mathbb{F}_p\to \mathbb{F}_p$ is a function then $f$ is polynomial.
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2answers
36 views

Why is every finitely generated field over $k$ not a finite type $k$-algebra?

A field extension $k\subseteq F$ is finitely generated if there exist $\alpha_a,\alpha_2,\dots,\alpha_n\in F$ such that $$F=k(\alpha_1)(\alpha_2)\dots(\alpha_n)$$ This is not the same as saying $F$ ...
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0answers
48 views

The Galois Field for the polynomial $x^3 - 2$.

I am reading a textbook prior to taking my first course in Field Theory. I think that if someone could answer the following 4 questions simply as True of False I might be less confused. I am denoting ...
3
votes
1answer
34 views

Show that certain matrices over rings form a field

I have got the following assignment: $R$ is a ring, $K:=\{ \begin{pmatrix} a & b \\ -b & a \\ \end{pmatrix}: a,b \in R\}$ I need to show that $K$ is a field. And I believe it is not ...
2
votes
1answer
121 views

Example of a chain without a supremum in a non Archimedean ordered field

I give here the example of a non-Archimedean ordered field. I know that the field is not order complete. What is a simple example in that field of a chain without a supremum?
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0answers
92 views

Discrete valuation fields and representation as power series

Let $(K,v)$ be a discrete valuation field ($v$ is surjective). Let $\mathcal O$ be the ring of integers of $v$ and moreover let $\mathfrak p$ be the unique maximal ideal of $\mathcal O$. Then we have ...
1
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1answer
36 views

Finding Galois conjugates

I'm working on a big exercise from Dummit & Foote (p.584) with the end goal of constructing a polynomial with Galois group $Q_8$ (Quaternion group of order $8$). Take $$\alpha = ...
4
votes
4answers
135 views

The equation $-1 = x^2 + y^2$ in finite fields

In an ordered field we have $x^2 \ge 0$, hence the equation $-1 = x^2 + y^2$ has no solution. But what about finite fields in general? What is the solutions set $$ -1 = x^2 + y^2 $$ of this equation? ...
0
votes
2answers
26 views

Splitting field of $x^2 +1 \;$ over $\mathbb Z_7 $

I need to find the splitting field of $\; x^2+1 \in \mathbb Z_7 [x] \;$ over $\mathbb Z_7 $. The roots of the polynomial are $-i \;$ and $i$. Therefore I would conclude that the splitting field is ...
0
votes
2answers
49 views

Finite Fields problem [closed]

"Given a Galois Field $(\mathbb{F}, +, \cdot)$ of order 8. With an element $x \in \mathbb{F}$ we create a group $(\{x^m | m \in \mathbb{Z}\}, \cdot)$. ($x^m$ is calculated via the second operator ...
2
votes
1answer
28 views

Product of degree of two field extensions of prime degree

Let $L/K$ be a field extension. Let be $\alpha, \beta \in \mathbb{C}$, such that $[\mathbb{Q}(\alpha):\mathbb{Q}] = p$, and $[\mathbb{Q}(\beta):\mathbb{Q}] = q$, for some prime numbers $p$ and $q$. ...
1
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1answer
31 views

Galois group of a polynomial over $\mathbb{C}[t]$

To find the Galois group of the polynomial $X^3-X-t\in\mathbb{C}[t]$, an approach is to compute the discriminant (equal $(2-\sqrt{27}t)(2+\sqrt{27}t)$) which is not a square in $\mathbb{C}[t]$ so the ...
0
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1answer
28 views

Invariant subfields and Galois group

Let $f$ be an irreducible polynomial of degree $n$ over $\mathbb{Q}$. Let $K$ be the splitting field of $f$ over $\mathbb{Q}$. Let $\alpha\in K$ be a root of $f$. Let $H$ be the subgroup of $G$ that ...
1
vote
1answer
45 views

Intersection of Kummer extension [closed]

Let $p$ and $q$ be two prime numbers and $\omega$ be the primitive 3rd root of unity. The splitting field of $X^3-p$ over $\mathbb{Q}$ is $K_p=\mathbb{Q}(p^{\frac{1}{3}},\omega)$ and we have a similar ...
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1answer
45 views

A Question on Integral domains and Fields [closed]

Suppose $F$ is just a non-zero commutative ring with a unit. I want to ask can we deduce that $F$ is an integral domain.
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3answers
21 views

In the finite field $F$ of characteristic $p$, is $a^{p^n} = a$?

If F is a finite field of characteristic $p$, $a$ is some element in $F$ and the number of elements in $F$ is $p^n$, is it true that $a^{p^n} = a$ for all $a$ in $F$? If it is, how could one prove or ...
0
votes
2answers
29 views

Show that a finite field of order $p^n$ has exactly one subfield of $p^m$ elements for each divisor $m$ of $n$.

Show that a finite field of order $p^n$ has exactly one subfield of $p^m$ elements for each divisor $m$ of $n$. Suppose that $o(F)=p^n$ .Let $F$ has $\Bbb Z_p$ as its prime subfield. Let $n=km$. I ...
0
votes
1answer
20 views

Prove that $\Bbb Z_2(\alpha)=\Bbb Z_2(\beta)$ .

Consider the polynomials $x^3+x^2+1,x^3+x+1$ over $\Bbb Z_2$ which have roots say $\alpha,\beta $ respectively. Prove that $\Bbb Z_2(\alpha)=\Bbb Z_2(\beta)$ . Since both $x^3+x^2+1,x^3+x+1$ are ...
0
votes
1answer
39 views

Prove that every finite extension field of $\Bbb R$ is either $\Bbb R$ itself or isomorphic to $\Bbb C$.

Prove that every finite extension field of $\Bbb R$ is either $\Bbb R$ itself or isomorphic to $\Bbb C$. I tried in this way.Let $E$ be a a finite extension of $\Bbb R$. Then $E$ is an ...
2
votes
1answer
23 views

Are the following options correct in case of a field?

I am reading field theory and i can't answer the following: 1.Is $\Bbb R$ algebraic over $\Bbb Q$? 2.If a field is algebraically closed then it has characteristic as $0$. Obviously $[\Bbb R:\Bbb ...
1
vote
1answer
28 views

Can someone please explain why it is the *smallest* subfield?

I am reading field theory and having trouble with: As Fraleigh writes: Let $E$ be an extension of $F$ .Define $\phi_\alpha:F[x]\to E;\phi_\alpha(a)=a;a\in F,\phi_\alpha(x)=\alpha$ . Suppose that ...
3
votes
1answer
41 views

Field extension whose tensor product with itself over $\mathbb{Q}$ is not a field

An old qual problem reads Let $D$ be a 9-dimensional central division algebra over $\mathbb{Q}$ and $K \subset D$ be a field extension of $\mathbb{Q}$ of degree $>1$. Show that $K ...
6
votes
2answers
408 views

Does the set of all fields exist ?

We often say "let F be a field", so I was wondering if we could consider, in ZFC, the set of all fields without some contradictions arising (so that we wouldn't have to use the global axiom of choice ...