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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Number of solutions for the given equation in finite field of order 32.

Let $F$ be a field of order 32. Find number of solutions $(x,y)\in F\times F$ for $x^2 +y^2 +xy =0$. I have figured out that non zero elements of this field forms a cyclic multiplicative group of ...
2
votes
1answer
29 views

Is $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$?

I'm trying to determine whether $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$, but I'm confused since $\mathbb{Z}/9\mathbb{Z}$ is not a field, but $x^2 + x + 1$ is irreducible in ...
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1answer
29 views

Transposition not diagonalizable in characteristic 2

In another thread it was proved that transposition as a linear map is diagonalizable. This, however, does not hold when we are working over a field of characteristic 2. I suppose the proof of this can ...
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0answers
26 views

Singular matrices over prime fields

Show that if a given matrix A with integer coefficients over $\mathbb{F}_{p}$ is singular for infinitely many primes $p$, then it is for all primes.
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1answer
25 views

A two-variable quadratic form over a field of characteristic 2 with no nontrivial roots

I'm looking for a quadratic form of the form $q(x,y)=ax^2 + bxy + cy^2 \in F[x,y]$, where $F$ has characteristic 2, and $q(x,y)$ has no roots besides the obvious one, $x=y=0$. I've proved the case of ...
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0answers
14 views

Switching blinding factors securely.

My question is related to information security area and I have asked almost a similar question in: ...
11
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3answers
718 views

Example of a category without product

A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category ...
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1answer
80 views

Algebraic element - integral domain

Let $K$ a field and $L$ a subfield of $K$. Let the set $\overline{L}:= \{k \in K: k$ is algebraic over $L$ $\}$ is another subfield of $K$. Show that $\overline{\overline{L}}=\overline{L}$. ...
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0answers
30 views

Regarding the number of isomorphisms between splitting fields

Let $\phi : F \rightarrow F_1$ be an isomophism of fields and $f(x) > \in F[x]$. Let $\Phi : F[x] \rightarrow F_1[x]$ be the unique ring isomorphism which extends $\phi$ and maps $x$ to ...
2
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0answers
20 views

How algebraically restrictive is the structure of $\Bbb Q_p$?

My question is a little difficult to explain, but I will try to state it. First, let me talk about the field $\Bbb R$ of real numbers. Let $K$ be the maximal real algebraic extension of $\Bbb Q$, that ...
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0answers
34 views

What does $F^n$ mean in the case of fields?

I am reading notes for Hilbert's Nullstellensatz and came across this expression. $$\exists F, \mathcal{V}_F(I):\{\xi \in F^n:f(\xi)=0, \forall f\in I\}\neq \emptyset$$ Here, $F=\mathcal{P}/m$ is a ...
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2answers
38 views

'Multiplying' by 0 in a field, field axiom proofs

The question says: The solution set was posted and there are a few things I don't quite understand from it. For the first one, I'm not entirely sure what's happening. It appears to be using the ...
2
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1answer
24 views

A question concerning archimedean places on number fields

Let $L$ be a number field and $G=Aut(L\mid \mathbb{Q})$. Let $|\cdot|$ be the usual archimedean value on $\mathbb{C}$ and, by abuse of notation, its restriction to $\mathbb{Q}$. Then the archimedean ...
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0answers
12 views

Linearly disjoint abelian extension of $\mathbb{Q}_p$

Given an abelian(finite or infinite) extension $K/\mathbb{Q}_p$ which is not the maximal abelian extension, can we always choose a cyclic extension $E=\mathbb{Q}_p(\zeta_n)$ such that $E\cap ...
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3answers
20 views

Show that $\{0\}$ and $V$ are the only linear subspaces of $V = k.$ where $k$ is a field.

On the surface this seemed easy, but my first attempt was rendered useless since I don't actually know that its an ordered field. I said: Let $W (\neq V)$ be a subspace of $ V$ then, if we let $v = ...
2
votes
1answer
30 views

Valuation ring between $F$ and $\mathcal O_F$

Let $(F,v)$ be a complete discrete valuation field (normalized) with ring of integers $\mathcal O_F$. Why cannot exist a valuation ring $A$ of $F$ such that $F\supsetneq A\supsetneq\mathcal O_F$ ...
2
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2answers
28 views

Suggestions for readings; Elliptic curves over function fields

I would love to know some good refercences about Elliptic curves over function fields. Especially in view with Mordell-Weil's Theorem. I am already familiar with the main proof of Mordell's theorem in ...
2
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1answer
25 views

Symmetric Polynomial in roots is in $F[X]$

I recently came across the following claim. Let $F$ be a field of characteristic $0$. Let $f\in F[X]$ have roots $y_1, \ldots , y_d$ in the algebraic closure of $F$. Define $$ g_h = \prod_{1\le ...
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3answers
27 views

Show that for $p \neq 2$ not every element in $\mathbb{Z}/p\mathbb{Z}$ is a square.

Show that for $p\neq2$, not every element in $\mathbb{Z}/p\mathbb{Z}$ is a square of an element in $\mathbb{Z}/p\mathbb{Z}$. (Hint: $1^2=(p-1)^2=1$. Deduce the desired conclusion by counting). So far ...
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1answer
24 views

Please find errors in my reasoning about field axioms

We can define a field F with the following properties: Binary operations + (addition) and ⋅ (multiplication) Commutativity Associativity Identities Inverses Distributivity Now, the additive ...
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2answers
26 views

Show that $K[X]/(P)$ is the splitting field of $P$.

Let $K$ a field and $P\in K[X]$ and irreducible polynomial. The fact that $K[X]/(P)$ is a field is fine. I want to show that it's the smallest field where that split $P$. First, let show that ...
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votes
2answers
60 views

Why does the associative property of vector addition imply a sum may be written as $\alpha_1+\alpha_2+\cdots+\alpha_n$?

In an effort to understand that a sum involving a number of vectors is independent of the way in which these vectors are associated, I've tried to derive other bindings of certain vector additions in ...
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1answer
20 views

Does the splitting field of an irreducible polynomial contain all extensions over which the polynomial factors?

Say $f$ is an irreducible polynomial with coefficients in a field $F$. Say $f$ is no longer irreducible over some extension $K$ of $f$, i.e. $f$ factors into a product of (irreducible) polynomials ...
2
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2answers
47 views

If a field element is simultaneously an $m$th and $n$th power… [closed]

Suppose $F$ is a field. Suppose we have $a,b \in F $ and relatively prime integers $m,n \geq 1$ such that $a^m = b^n$. Can I conclude that there is some $c \in F $ such that $c^{mn} = a^m = b^n$?
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0answers
37 views

Algebraic extension of perfect field is algebraically closed

Let $F$ be a perfect field, i.e. every irreducible polynomial over $F$ has distinct roots in the algebraic closure of $F$. Suppose that $K$ is an algebraic extension of $F$ with the property that ...
0
votes
1answer
40 views

Proof of equality of $2$ polynomials without using Galois Theory?

I have the following situation and ask myself whether an argument is available to prove the equality of the $2$ following polynomials. Both polynomials are of degree $n$ and are monic. The ...
4
votes
1answer
55 views

Algebraic closure with no nontrival automorphism

In Milne's notes on Galois theory, Chapter 7, p.91 he remarked that it is consistent without the axiom of choice that there exists an algebraic closure $L$ of $\mathbb{Q}$ with no nontrivial ...
1
vote
1answer
27 views

If $g(x) \in K[x]$, the $g(\alpha)=0$ if and only if $f(x)|g(x)$

Let $K $ a subfield of $\mathbb{C}$, $\alpha$ a complexe number which is algebraic on $K$ and $f(x) \in K[x]$ the minimal polynomial of $\alpha$ on $K$. If $g(x) \in K[x]$, the $g(\alpha)=0$ if ...
2
votes
1answer
43 views

Let $K$ a subfield of $\mathbb{C}$. Show that $\mathbb{Q} \subset K$

Let $K$ a subfield of $\mathbb{C}$. Show that $\mathbb{Q} \subset K$. To do this, I am trying to use an exercise done in class : Let $K$ a field, $\alpha \in K$ and $L$ a subfield of $K$. Then ...
2
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0answers
29 views

If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?

If $\alpha$ is an algebraic element and $L \subset K$ are both field, does the polynomial ring $L[\alpha]$ is also a field? I am trying to prove that the ring of fraction $L(\alpha)$ is equal to ...
1
vote
1answer
22 views

Can I find a Galois extension which contains a finite set of algebraic elements?

Suppose $K/F$ is an algebraic extension of fields. Pick a finite collection $a_1,...,a_n \in K$. Can I find a Galois extension of $F$ which contains these $n$ elements of $K$?
2
votes
1answer
40 views

Suppose $\phi\in{Aut(C)}$ and continuous, why $\phi$ must fix $R$? [closed]

Suppose $\phi\in{Aut(\mathbb{C})}$ and continuous, why $\phi$ must fix $\mathbb{R}$? I know that continuity is crucially important here, but I'm not sure how.
1
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1answer
46 views

Are the ordered field axioms consistent?

Today in class a student asked to the professor "Are the ordered field axioms consistent?" And my prof replied something along the lines of "Yes, as we have a model of them: $\Bbb R$, this ...
1
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2answers
37 views

Simple Proof for Commutative Property of Multiplication

I'm supposed to show that $a\cdot b=b\cdot a$ for a set $K:=\{s+t\sqrt2:s,t\in\mathbb{Q}\}$ to show that this set is a field. I was going to set it up like: Let $a, b\in K$ such that ...
0
votes
1answer
25 views

Imposing ordering on $Q(\mathbb{R}[x])$

I'm working on a question that asks to show that the field of fractions of the polynomial ring with real coefficients is an ordered field by defining $$\frac{p}{q} > 0 \iff \frac{a_m}{b_n} > 0 ...
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0answers
21 views

how many orbits are possible in the group action?

Let G be Galois group of a field with nine elements over its Subfield with three elements.Then the number of orbits for action of G on the field with nine elements?
0
votes
1answer
27 views

Proof of $ \forall a \in \Bbb{R}: -a = (-1) a $.

Problem. Prove that $ \forall a \in \Bbb{R}: -a = (-1) a $. I already have a proof but I would like to see another. :) Proof by contradiction Suppose that $ -a \neq (-1) a $. Then ...
1
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1answer
26 views

Relation of F and V in a Vector Space

In many books and on Wikipedia a vector space is defined as a tuple $(F, +, V)$ where $F$ is a field and $V$ an abelian Group plus some axiums that must hold which I will omit here. I also often see ...
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0answers
34 views

Splitting field.

Show that $\mathbb {Q }(\pi)$ is not splitting field over $\mathbb {Q }(\pi^2)$. I am thinking $\mathbb {Q }(\pi)$ and $\mathbb {Q }(\pi^2)$ are same field Or $\mathbb {Q }(\pi)$ is not even a ...
0
votes
1answer
46 views

Associative algebra over the field of real numbers

Prove that the associative algebra without divisors of zero over the field $\mathbb R$ of dimension greater than 2 can not be commutative. I'm new to this and I will be very grateful for your help!
4
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2answers
76 views

Is there only one way to make $\mathbb R^2$ a field?

I think I read an answer to this question before but I can't find it by searching. We can make $\mathbb R^2$ a field by defining addition as normal and defining multiplication by complex ...
1
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1answer
30 views

Constructing infinite field in which all subrings are subfields

A classmate posed a question in class as to if there existed an infinite field $F$ for which every subring $R \subseteq F$ was a subfield. We'd already determined that if $F$ was a finite field, then ...
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1answer
43 views

field properties from prime subfield

Given a field $F_{p^n}$ with $char(F)=p$ and $p$ prime. And thus our main misunderstanding: Why is the arithmetic in $F_{p^n}$ modular p? Why is it that it's prime field $F_{p}$ forces upon $F_{p^n}$ ...
2
votes
1answer
42 views

What do the square bracket signify in $\int [\text{d}\pi]f(\pi)$

I am reading this paper which repeatedly includes integrals such as, $$ P_M(\phi \to \phi') = \int [\text{d}\pi][\text{d}\pi'] P_G(\pi)\delta((\phi, \pi) - (\phi'', \pi'')) $$ Note ...
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vote
3answers
81 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
33 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= ...
0
votes
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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votes
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 $J\subseteq I$.
2
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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 ...