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 ...
32
votes
4answers
1k views
What kind of work do modern day algebraists do?
Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
30
votes
2answers
371 views
Is it true in an arbitrary field that $-1$ is a sum of two squares iff it is a sum of three squares?
Here's a statement from Lam's First Course in Noncommutative Rings. (Paraphrased)
Let $k$ be a field. Then the following conditions are equivalent. $$(\forall a,b,c,d\in k)\;\;(a,b,c,d)\neq ...
28
votes
2answers
2k views
The square roots of the primes are linearly independent over the field of rationals
I need to find a way of proving that the square roots of a finite set
of different primes are linearly independent over the field of
rationals. I've tried to solve the problem using elementary ...
28
votes
5answers
1k views
Continuity of the roots of a polynomial in terms of its coefficients
It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
25
votes
4answers
736 views
Fermat's Last Theorem and Kummer's Objection
In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
25
votes
3answers
372 views
Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”?
The function
$f(x)=x+\sin(x)$
is easily checked to be a bijection from the reals to itself, and so it has a unique inverse $y\mapsto g(y)$ such that $f\circ g=g\circ f$ are both the identity map.
...
21
votes
6answers
535 views
Is $\mathbf{C}$ the algebraic closure of any field other than $\mathbf{R}$?
It seems to me (intuitively) that there should be no other fields whose algebraic closure is $\mathbf{C}$, even though I have no reason to believe it. The facts I've been using to formulate an ...
21
votes
3answers
437 views
On the meaning of being algebraically closed
The definition of algebraic number is that $\alpha$ is an algebraic number if there is a nonzero polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $p(\alpha)=0$.
By algebraic closure, every nonconstant ...
20
votes
6answers
565 views
Axiomatic characterization of the rational numbers
We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this).
...
19
votes
1answer
2k views
Example of infinite field of characteristic $p\neq 0$
Can you give me an example of infinite field of characteristic $p\neq0$?
Thanks.
19
votes
1answer
282 views
When, and by whom, was “$\mathbb{C}$ is algebraically closed” dubbed the “fundamental theorem of algebra”?
Wikipedia has this enigmatic sentence on the page for the fundamental theorem of algebra:
...its name was given at a time when the study of algebra was mainly concerned with the solutions of ...
18
votes
2answers
362 views
What is the coproduct of fields, when it exists?
This is a slightly more advanced version of another question here.
Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by ...
18
votes
2answers
440 views
field generated by a set
Let $S$ be the set of real numbers which can be written in the form $ \sum_{n\geq0}{ \frac{\epsilon_{n}}{n!}}$ ,where ${\epsilon_n}^2=\epsilon_n$ and let $K$ be the field generated by $S$ , help me ...
17
votes
6answers
794 views
Why should I care about fields of positive characteristic?
This is what I know about why someone might care about fields of positive characteristic:
they are useful for number theory
in algebraic geometry, a theory of "geometry" can be developed over them, ...
17
votes
1answer
870 views
Is there a purely algebraic proof of the Fundamental Theorem of Algebra?
Among the many techniques available at our disposal to prove FTA, is there any purely algebraic proof of the theorem?
That seems reasonably unexpected, because somehow or the other we are depending ...
16
votes
2answers
479 views
Why algebraic closures?
Let me begin by summarizing the question:
Why do we care about fields closed under rational exponentiation, and less about fields closed under other operations?
Historically the solution for ...
16
votes
1answer
280 views
Irreducibility of $x^{n}+x+1$
Motivated by this problem, and KCd's comment on my answer, I am left with the following question:
Question: Suppose that $n\not \equiv 2\pmod{3}$. Is $$x^n+x+1$$ irreducible over $\mathbb{Q}$?
...
15
votes
2answers
606 views
Why is $\mathbb{Q}(t,\sqrt{t^3-t})$ not a purely transcendental extension of $\mathbb{Q}$?
This question is taken from Dummit and Foote(14.9 #6). Any help will be appreciated:
"Show that if $t$ is transcendental over $\mathbb{Q}$, then $\mathbb{Q}(t,\sqrt{t^3-t})$ is not a purely ...
14
votes
5answers
404 views
Why does $K \leadsto K(X)$ preserve the degree of field extensions?
The following is a problem in an algebra textbook, probably a well-known fact, but I just don't know how to Google it.
Let $K/k$ be a finite field extension. Then $K(X)/k(X)$ is also finite with ...
14
votes
1answer
360 views
An exercise with Zariski topology
I read this exercise:
Prove that the set $S = \{ (n, 2^n, 3^n ) \mid n \in \mathbb{N} \}$ is dense in $\mathbb{C}^3$ with Zariski topology.
I have seriously thought about it, but I do not manage to ...
14
votes
3answers
262 views
Finding the degree of a field extension over the rationals
Let $\alpha = \sqrt{\sqrt{2}+\root 4 \of 2}$, $\beta =\sqrt{\sqrt{2}- \root 4 \of 2}$, $\gamma = \sqrt{-\sqrt{2}+i\root 4 \of 2}$ and $\delta = \overline{\gamma}=\sqrt{-\sqrt{2}-i\root 4 \of 2}$.
Let ...
14
votes
4answers
2k views
Finding the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$.
I have to find the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$. The suggested way of doing it is to prove that $\mathbb Q[\sqrt 2 + \sqrt[3] 2]=\mathbb Q[\sqrt 2,\sqrt[3] 2]$ first.
...
14
votes
5answers
599 views
Why isn't the perfect closure separable?
Let $F\subset K$ be an algebraic extension of fields. By taking the separable closure $K_s$, we obtain a tower $F\subset K_s \subset K$ such that $F\subset K_s$ is separable and $K_s\subset K$ is ...
13
votes
3answers
2k views
Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$?
Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$ ?
$$\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbf{Q}\}$$
...
13
votes
2answers
416 views
Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$
Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
13
votes
4answers
492 views
How to prove that the sum and product of two algebraic numbers is algebraic?
Suppose $E/F$ is a field extension and $\alpha, \beta \in E$ are algebraic over $F$. Then it is not too hard to see that when $\alpha$ is nonzero, $1/\alpha$ is also algebraic. If $a_0 + a_1\alpha + ...
13
votes
2answers
374 views
Is it actually incorrect to say $x/1 = x$?
The rational numbers $\mathbb{Q}$ are defined as the field of quotients of $\mathbb{Z}$ under the relation $(a, b) \sim (c , d) \iff$ $ad = bc$. There is an obvious isomorphism between the subring ...
13
votes
2answers
604 views
Basis of primitive nth Roots in a Cyclotomic Extension?
While reading one of Keith Conrad's great blurbs, Linear Independence of Characters, there is a footnote at the bottom of page 2 saying
In general, the primitive $n$th roots of unity in the $n$th ...
13
votes
1answer
306 views
Why is $\mathbb{C}_p$ isomorphic to $\mathbb{C}$?
I know that two closed fields of caracteristic $0$ and uncountable are isomorphic iff they have the same cardinality. But I don't know why $\mathbb{C}_p$ has the same cardinality as $\mathbb{C}$. Can ...
13
votes
2answers
306 views
Atiyah's definitions of Topological Quantum Field Theory
According to Atiyah, a TQFT is a functor from the category of cobordisms to the category of vector spaces.
How does this definition relate with the physics of quantum mechanics?
What does the ...
13
votes
3answers
239 views
Is the Pythagorean closure of $\mathbb Q$ equal to the field of constructible numbers?
A Pythagorean field is one in which every sum of two squares is again a square. $\mathbb Q$ is not Pythagorean, which is easy to see. I have read a theorem online which says that every field has a ...
13
votes
3answers
382 views
Is $\bar{\mathbb{Q}}(x)\cap \mathbb{Q}((x))=\mathbb{Q}(x)$? [unsolved (even though we earlier thought it was)]
Fix the algebraic closure of $\mathbb{Q}((x))$ for this question to make sense. I know that $\mathbb{Q}((x)) \cap \overline{\mathbb{Q}(x)}$ has elements that are not in $\mathbb{Q}(x)$ (in analogy to ...
12
votes
4answers
624 views
$\mathbb R^3$ is not a field
I'm trying to prove that $\mathbb R^3$ is not a field with component-wise multiplication and sum defined. I think it's weird, because every properties of a field are inherit from $\mathbb R$.
Anyone ...
12
votes
3answers
361 views
Are $\mathbb{R}$ and $\mathbb{Q}$ the only nontrivial subfields of $\mathbb{R}$?
I've been asked to prove that any subfield of $\mathbb{R}$ contains $\mathbb{Q}$, and I know how to do it, but it made me wonder if there were subfields of $\mathbb{R}$ that strictly contained ...
12
votes
2answers
214 views
A sentence false in a field of characteristic $0$ but true in all fields of positive characteristic?
Consider the language $L_{Ri}=\{+,\cdot, 0, 1\}$ of rings. It is easy to show using compactness that if $\sigma$ is a sentence that holds in all fields of characteristic $0$, there is some $N\in ...
11
votes
3answers
337 views
Why is the difference of distinct roots of irreducible $f(x)\in\mathbb{Q}[x]$ never rational?
The way I understand it, is that if $f(x)$ is an irreducible polynomial in $\mathbb{Q}[x]$ of degree at least 2, then a difference of distinct roots $a_i-a_j$ is never rational for any of the ...
11
votes
1answer
261 views
Field reductions
If there is a field $F$ that is a field reduction of the real numbers, that is $F(a)=\mathbb{R}$ for some $a$, let's also denote this $F=\mathbb{R}(\setminus a)$, then given $x \in \mathbb{R}$ is ...
11
votes
2answers
114 views
Is a bivariate function that is a polynomial function with respect to each variable necessarily a bivariate polynomial?
Let $ \mathbb{F} $ be an uncountable field. Suppose that $ f: \mathbb{F}^{2} \rightarrow \mathbb{F} $ satisfies the following two properties:
For each $ x \in \mathbb{F} $, the function $ ...
11
votes
3answers
144 views
Fundamental Theorem of Algebra for fields other than $\Bbb{C}$, or how much does the Fundamental Theorem of Algebra depend on topology and analysis?
When proving the Fundamental Theorem of Algebra, we need to appeal to analytic and/or topological properties of $\Bbb{C}$ and $\Bbb{C}[z]$. Is this going to be necessary in general, and if so, to what ...
11
votes
2answers
742 views
Grassmann numbers as eigenvalues of nilpotent operators?
The following question is motivated by the construction of the fermionic path/field integral, as done for example in Altland & Simons "Condensed Matter Field Theory".
Consider the vector space ...
10
votes
10answers
1k views
Proving $\sqrt 3$ is irrational.
There is a very simple proof by means of divisivility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum:
Suppose
...
10
votes
6answers
842 views
What is a field?
I've always wondered about what a field is meant to represent. For example, group automorphisns naturally represent symmetry in many areas.
I'm not looking for a solid answer, just an idea.
10
votes
3answers
409 views
Why doesn't stuff hold in characteristic non-zero?
There are a bunch of theorems in algebra that require the underlying field to be characteristic 0. I seem to remember that these all stemmed from one basic fundamental theorem that only holds in ...
10
votes
3answers
506 views
A question regarding the definition of Galois group
In my book, Galois group is defined to mean the set of automorphisms on $E/F$ that "leave alone" the elements in $F$.
On Wikipedia it says:
"If $E/F$ is a Galois extension, then $Aut(E/F)$ is called ...
10
votes
2answers
181 views
Can all polynomials of a given degree be reducible?
Let $n > 1$ be a fixed integer. Does there exist a field $F$ with the following properties?
$F$ is not algebraically closed.
Every polynomial $f(x) \in F[X]$ of degree $n$ is reducible.
I ...
10
votes
2answers
169 views
Is it possible to construct an ordered field which is also algebraically closed?
It is well known that while the real numbers are totally ordered, they are not algebraically closed, and while the complex numbers are algebraically closed, they are not totally ordered. Is it ...
10
votes
2answers
622 views
Luroth's Theorem
I have just begun to read Shafarevich's Basic Algebraic Geometry. In the first section of the first chapter, he quotes Luroth's theorem, which states that any subfield of $k(x)$ that is not just $k$ ...
10
votes
1answer
81 views
Other Euler characteristics?
At the end of V.3.4 in Algebra: Chapter 0, Aluffi describes the construction of a Grothendieck group over the category of finite dimensional $\operatorname{k}$-vector spaces, ...
10
votes
2answers
283 views
Analog to the primitive element theorem for transcendental extensions?
The Primitive Element Theorem states that if $E/F$ is a finite separable field extension, then there exists an element $a$ such that $E=F(a)$.
There's a similar result I found, that I don't quite ...
9
votes
3answers
653 views
Galois Group of $(x^3-5)(x^2-3)$
I am having some trouble calculating the Galois group (over $\mathbb{Q}$) of $(x^3-5)(x^2-3)$. I can see the splitting field is ...
