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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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
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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votes
2answers
439 views

Are all existence proofs by contradiction?

Do all existence proofs proceed indirectly (a.k.a "by contradiction," "reductio ad absurdum," etc.)? For example, can we prove directly that in a field the additive identity element and the ...
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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 ...
24
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2answers
867 views

Do the Liouville Numbers form a field?

The Liouville numbers are those which are better-than-polynomially approximated by rationals. More precisely, we say $x\in\mathbb{R}$ is Liouville when for all $n\in\mathbb{N}$ there is a $\tfrac ...
0
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2answers
68 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 ...
1
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1answer
21 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 ...
0
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1answer
41 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 ...
2
votes
1answer
41 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.
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0answers
38 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 ...
2
votes
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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1answer
58 views

Does Hilbert 90 need the extension to be Galois?

I read about Hilbert 90 in Morandi's Fields and Galois Theory (GTM 167). Let $K/F$ be a finite Galois extension. Let $G=\operatorname{Gal}(K/F)$. Define a crossed homomorphism $f\colon G\to K^*$ ...
2
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0answers
30 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 ...
4
votes
1answer
56 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 ...
2
votes
5answers
309 views

how to prove $(-1)\cdot(-1)=1$ based only on the field axioms?

How do I prove $(-1)\cdot(-1)=1$ (or in other words that $-1$ is the multiplicative inverse of itself), based only on the field axioms? Thanks
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1answer
28 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 ...
1
vote
1answer
23 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
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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 ...
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 ...
9
votes
3answers
1k views

Understanding the characteristic of a field

The characteristic of a field is defined to be the smallest positive integer $p$ such that $$p \cdot 1 = 0.$$ But I have learned that field has no zero divisors. How is this possible?
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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 ...
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
vote
3answers
86 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 $\mathbb{Q}$) ...
1
vote
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
36 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
47 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
votes
2answers
77 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
vote
1answer
31 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 ...
0
votes
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 ...
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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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 $J\subseteq I$.
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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) &= ...
2
votes
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 ...
0
votes
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
vote
2answers
32 views

Finite extensions of $\mathbb F_p(t)$ [on hold]

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 ...
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 ...
1
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0answers
33 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
votes
2answers
46 views

Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable?

Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable ? I would say yes since the fact that $x$ separable over $F$ implies $E(x)/F$ separable, an since $E=E(x)$ then $E/F$ ...
0
votes
1answer
29 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 ...
2
votes
0answers
69 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 ...
0
votes
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
vote
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 ...
2
votes
2answers
54 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 ...