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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15
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1answer
325 views

Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that: There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$ Can someone provide an ...
15
votes
2answers
328 views

Sum of irrational numbers, a basic algebra problem

Let $x_1,\dots,x_n$ be positive rational numbers. If $\sqrt[l_1]{x_1},\dots,\sqrt[l_n]{x_n}$ are all irrational numbers (where $l_1,l_2,\dotsc,l_n\in\Bbb N^*$), does it follow that $$\sqrt[l_1]{x_1}+ ...
15
votes
1answer
359 views

A *finite* first order theory whose finite models are exactly the $\Bbb F_p$?

Since this question turned out to be trivial, I'm now asking this strengthened version: Is there a finitely axiomatized first order theory $T$ in the language of rings such that its finite models ...
14
votes
9answers
404 views

Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible

Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it divides -4 and the constant coefficient 16, don't ...
14
votes
3answers
851 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 ...
14
votes
2answers
2k views

Proving the “freshman's dream”

$(x+y)^p = x^p + y^p$ holds in any field of characteristic $p$. However all the proofs I have seen use induction and some relatively nasty algebra despite how fundamental this fact seems. What is the ...
14
votes
5answers
453 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
4answers
863 views

Proving $\sqrt{a+\sqrt{b}}=\sqrt{m}+\sqrt{n}\iff a^{2}-b$ is a square

This is an exercise for the book Abstract Algebra by Dummit and Foote (pg. 530): Let $F$ be a field of characteristic $\neq2$ . Let $a,b\in F$ with $b$ not a square in $F$. Prove ...
14
votes
2answers
405 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 ...
14
votes
4answers
5k 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
3answers
332 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 ...
14
votes
3answers
411 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 ...
13
votes
6answers
1k 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.
13
votes
3answers
479 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 ...
13
votes
2answers
335 views

A sentence false in a field of characteristic $0$ but true in all fields of positive characteristic?

Consider the language $L=\{+,\cdot, 0, 1\}$ 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 $N\in \mathbb N$ ...
13
votes
4answers
466 views

A field of order $32$

I was working on this problem from an old qual exam and here is the question. In particular this is not for homework. True or False: There are no fields of order 32. Justify your answer. ...
13
votes
3answers
2k views

Inseparable, irreducible polynomials

The standard examples of irreducible, inseparable polynomials that one encounters in an introductory course on field theory all seem to have only a single root in an algebraic closure. Are there ...
13
votes
3answers
585 views

Characterization of a subfield $K \varsubsetneq \mathbb {C}$ and $x\in \mathbb{R}$

Characterize $x \in \mathbb R$ such that there exist a subfield $K \varsubsetneq \mathbb C$ such that $K(x) = \mathbb C$. -All subfields $K$ of $\mathbb{C}$ contain $\mathbb Q$, then all $x\notin ...
13
votes
3answers
233 views

Can we construct $\Bbb C$ without first identifying $\Bbb R$?

Sometimes it is useful to consider $\Bbb C$ as our primitive and identify $\Bbb R$ as a subset of $\Bbb C$. Thus we can define $\Bbb R$ (or at least a set with all of the interesting properties of ...
13
votes
1answer
354 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
3answers
437 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 ...
13
votes
1answer
310 views

Families of Polynomials Irreducible in $\mathbb{Z}$ but reducible in $\mathbb{Z}/p\mathbb{Z}$ for all primes $p$.

I am wondering if there exist classification of polynomials that are irreducible in $ \mathbb{Z}$ but reducible $\pmod p$ for all primes $p$. I am aware that $\Phi_n$ has this property if ...
12
votes
3answers
9k views

what is difference between a ring and a field

The ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition. A field can be thought of as two groups with extra ...
12
votes
2answers
536 views

Is number rational?

How can we check if number $a=\frac{ \sqrt[4]{2}+\sqrt[3]{3}}{\sqrt[4]{2}+\sqrt[3]{3} +1}$ is rational? Is there any smart solution? Another assignment is to find $\left( ...
12
votes
3answers
704 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 ...
12
votes
2answers
227 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 ...
12
votes
2answers
306 views

Is $\mathbb Q(\sqrt{2},\sqrt{3},\sqrt{5})=\mathbb Q(\sqrt{2}+\sqrt{3}+\sqrt{5})$.

Is $\mathbf Q(\sqrt 2,\sqrt 3,\sqrt 5)=\mathbf Q(\sqrt 2+\sqrt 3+\sqrt 5)$? Say $L=\mathbf Q(\sqrt 2,\sqrt 3,\sqrt 5)$ and $K=\mathbf Q(\sqrt 2+\sqrt 3+\sqrt 5)$. It is easy to show that ...
12
votes
2answers
618 views

Why are there so many universal properties in math?

I don't really understand why there are so many universal properties in math or why they all need to be highlighted. For example, I'm studying some Algebra right now. I have found three universal ...
12
votes
1answer
126 views

Field Extensions and their dimensions

There is a powerful theorem with respect to a field $F$ extensions and their dimensions. $F<E<K \ \Rightarrow [K:F] = [K:E][E:F] $ This is analogous to the famous Lagrange's theorem with ...
12
votes
1answer
286 views

Is a field perfect iff the primitive element theorem holds for all extensions, and what about function fields

Let $L/K$ be a finite separable extension of fields. Then we have the primitive element theorem, i.e., there exists an $x$ in $L$ such that $L=K(x)$. In particular, the primitive element theorem ...
12
votes
1answer
750 views

Galois group of a reducible polynomial over $\mathbb {Q}$

Let $f \in\mathbb {Q}[X]$ be reducible - for the sake of simplicity, $ f = gh$ with $g,h \in\mathbb {Q}[X]$ irreducible. Let L be the splitting field of f. Does $Gal(f) \simeq Gal(g) \times Gal(h)$ ...
12
votes
2answers
862 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 ...
12
votes
1answer
502 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
12
votes
3answers
1k views

Recognizing when a tower of Galois extensions gives a Galois extension

For the easiest case, assume that $L/E$ is Galois and $E/K$ is Galois. Under what conditions can we conclude that $L/K$ is Galois? I guess the general case can be a bit tricky, but are there some ...
11
votes
4answers
1k 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 ...
11
votes
3answers
528 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 ...
11
votes
3answers
1k views

How to solve polynomial equations in a field and/or in a ring?

I'm studying for my exam, and I stuck on solving polynomials in a field and/or in a ring. Let me give you some examples: (1) Solve equation $x^2+4x+3=0$ in field $\mathbb{Z}_5$, $\mathbb{Z}_8$ and in ...
11
votes
1answer
1k views

Constructive proof of the existence of an algebraic closure

It is well-known that, assuming the axiom of the choice (in the form of Zorn's lemma), one can prove that any field $F$ has an algebraic closure. One proof roughly goes as follows: consider the ...
11
votes
3answers
2k views

Every algebraic extension of a perfect field is separable and perfect

I am trying to prove this statement in the characteristic $p>0$ case. Every algebraic extension of a perfect field is separable and perfect. This is stated as a corollary of Proposition ...
11
votes
2answers
2k views

Why does an irreducible polynomial split into irreducible factors of equal degree over a Galois extension?

I've been struggling to prove this fact over the past day or so. Suppose $f(x)\in F[X]$ is irreducible over a field $F$ with $\deg(f)=n$, and let $L$ be the splitting field of $f(x)$ over $F$ with ...
11
votes
1answer
183 views

Are the real numbers a nontrivial simple extension of another field?

Is there a proper subfield $K$ of the real numbers and a real number $\theta$ such that $\mathbb R = K(\theta)$? I thought of this question earlier idly wondering about what the structure of the ...
11
votes
3answers
455 views

Why is the collection of all algebraic extensions of F not a set?

When proving that every field has an algebraic closure, you have to be careful. In this article https://proofwiki.org/wiki/Field_has_Algebraic_Closure, and as I have been told on this site, if we have ...
11
votes
1answer
286 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
1answer
113 views

Subfield of $\mathbb{R}$ such that $\Bbb R/K$ is finite.

Is there a field $K \subset \mathbb{R}$ such that $1 < [\mathbb{R} : K] < \infty$? i.e a proper subfield of $\mathbb{R}$ such that the field extension $\mathbb{R}/K$ is finite.
11
votes
2answers
1k 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 $k$ ...
11
votes
2answers
218 views

Galois Group of $x^{4}+7$

I am trying to find the Galois group of $x^{4}+7$ over $\mathbb{Q}$ and all of the intermediate extensions between $\mathbb{Q}$ and the splitting field of this polynomial. I have found that the ...
11
votes
2answers
222 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
2answers
344 views

Is there a geometric interpretation of $F_p,\ F_{p^n}$ and $\overline{F_p}?$

I have been doing some exercises about finite fields lately and I think I've obtained some understanding of what they are. What seems to be missing though is some kind of picture. Learning to work ...
11
votes
2answers
157 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
11
votes
2answers
132 views

Infinite direct product of fields.

Let $F$ be a field, and consider the infinite direct product$$F \times F \times F \times F \times \dots,$$i.e. $\prod_{i=0}^\infty F$, i.e. the direct product of a countable number of copies of $F$. ...