Title: Theory / Important / Math / Al gebra / Abstract Algebra - Dummit - Foote+14{ / 4 Field
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On every note of this deck, unless a note says otherwise:
Chapter: 13
Chapter: 13

Name: A field
Definition: A commutative division ring.
Page: 510, 263, 34
Chapter: 13, 7, 1
Section: 1, 5, 4
Index: , 5
Kind: , Ex
Definition2: A ring \((F,+,\times)\) such that \((F^\times,\times)\) is an Abelian group.
Definition3: An integral domain \(R\) whose field of fractions is \(R\)
Definition4: A pair of rings, such that one is distributive over the other

Definition: \(f+\dots+f\), with \(n\) copy of \(f\). Or \(0_F\) if \(n=0\).
Page: 510
Chapter: 13
Section: 1
Variables: \(F\) a field,
\(n\in\mathbb N\),
\(f\in F\)
Notation: \(n\cdot f\)

Name: The characteristic of \(F\)
Definition: minimal such that \(n\cdot 1_F=0\) if it exists, \(0\) otherwise.
Page: 510
Chapter: 13
Section: 1
Variables: \(F\) a field
Notation: \(\text{ch}(F)\)
Index: , 1
Kind: Def, Prop
DefType: A number \(n\in\mathbb N\), 0 or prime

Question: \(p\cdot F\)
Answer: \(=0\).
Page: 510
Section: 1
Variables: \(F\) a field,
\(p\) its characteristic,
\(f\in F\)

Question: The characteristic of \(\mathbb R\) is
Answer: 0.
Page: 510
Section: 1
Index: (1)
Kind: Ex

Question: \(\text{ch}(\mathbb Z)=\)
Answer: \(0\).
Page: 510
Section: 1
Index: (1)
Kind: Ex

Question: \(\text{ch}(\mathbb Q)=\)
Answer: \(0\).
Page: 510
Section: 1
Index: (1)
Kind: Ex

Question: \(\text{ch}(\mathbb F_p)=\)
Answer: \(p\).
Page: 510
Section: 1
Variables: \(p\) prime
Index: (2)
Kind: Ex

Question: \(\text{ch}(\mathbb F_p[x])=\)
Answer: \(p\).
Page: 510
Section: 1
Variables: \(p\) prime
Index: (2)
Kind: Ex

Question: \(\text{ch}(\mathbb F_p(x))=\)
Answer: \(p\).
Page: 510
Section: 1
Variables: \(p\) prime
Index: (2)
Kind: Ex

Definition: \(-(n\cdot f)\)
Page: 511
Chapter: 13
Section: 1
Variables: \(F\) a field,
\(n\in\mathbb N\),
\(f\in F\)
Notation: \((-n)\cdot f\)

Definition: \(p\cdot 1_F=0\)
Page: 511
Chapter: 13
Section: 1
Variables: \(F\) a field,
\(p\in\mathbb N\) prime
Notation: \(p=0\)
Kind: Rem

Name: An extension field of \(F\)
Definition: A field \(K\) such that \(F\) is a subfield of \(K\).
Page: 511, 512
Chapter: 13
Section: 1
Variables: \(F\) a field
Kind: Def
Name2: An extension of \(F\)
DefType: A \(F\) vector space

Name: \(K\) over \(F\)
Definition: \(K\) is an extension of \(F\)
Page: 511, 512
Chapter: 13
Section: 1
Variables: \(F,K\) fields,
Notation: \(K/F\)
Kind: Def

Name: The base field of the extension
Definition: \(F\)
Page: 511
Chapter: 13
Section: 1
Variables: \(F\) a field
\(K\) an extension of \(F\)
Context: Field extension
Kind: Def

Name: The degree of the extension
Definition: The degree of \(K\) as an \(F\)-vector space
Page: 512
Chapter: 13
Section: 1
Variables: \(F\) a field
\(K\) an extension of \(F\)
Notation: \([K:F]\)
Example: infinity
Name2: The relative degree of the extension
Example2: \(n\)
Example3: 2
Applied to: \([\mathbb R:\mathbb Q]\)
Name3: The index of the extension
Applied to3: \([\mathbb C:\mathbb R]\)
Applied to2: \([\mathbb F_{p^n}:\mathbb F_p]\)

Name: A finite extension
Definition: An extension whose degree is finite
Page: 512
Chapter: 13
Section: 1
Variables: \(F\) a field

Question: \(\phi\) is
Answer: \(0\) or injective.
Page: 512
Section: 1
Variables: \(F,F'\) a field,
\(\phi:F\to F'\) an homomorphism of fields
Index: 2
Kind: Prop

Smaller0: \(p\) irreducible
Greater0: \(F[x]/(p(x))\)
Page: 512, 514
Section: 1
Variables: \(F\) a field,
\(p(x)\in F[x]\)
Index: 3
Kind: Theo
Hide middle: yes
Greater1: \(p\) has a root
Connect Greater0: is an extension of \(F\) such that
Greater2: a field
Connect Greater1: and is

Question: \([F[x]/(p(x)): F]=\)
Answer: \(\text{deg }p \)
Page: 513
Section: 1
Variables: \(F\) a field,
\(p[x]\in F[x]\), irreduccible
Index: 4
Kind: Theo

Smaller0: \(\mathbb R[x]\)
Greater0: \(\mathbb C\)
Page: 515
Section: 1
Index: (1)
Kind: Ex
Smaller to greater: is isomorphic to
Hide middle: yes
Smaller1: \((x^2+1)\)
Connect Smaller0: \(/\)

Q: \([...]/[...]\) is isomorphic to [...].
A: \mathbb Q[x]; (x^2+1); the field \(\{a+ib\mid a,b\in\mathbb Q\}\)
Page: 515
Chapter: 13
Section: 1
Index: (2)
Kind: Ex

Smaller0: \(\theta^{-1}\)
Greater0: \(-\frac1{p_0}\left(\sum_{i=1}^n p_i\theta^{i-1}\right)\)
Page: 516
Section: 1
Variables: \(F\) a field,
\(p(x)=\sum_{i=0}^{n} p_ix^i\in F[x]\),
\(\theta\) a root of \(p\)
Index: (5)
Kind: Ex
Smaller to greater: is equal to
Hide middle: yes
Assuming: \(p\) irreducible
Context: \(\mathbb F/(p(x))\)

Question: \(\mathbb F_2/(x^2+x+1)\) is
Answer: \(\{a+b\theta\mid a,b\in \mathbb F_2\}\),
\(((a+b\theta),(a'+b'\theta))\mapsto((a+a')+(b+b')\theta)\),
\(((a+b\theta),(a'+b'\theta))\mapsto((ab'+a'b)+(ab'+ba'+a'b')\theta)\)
Page: 516
Section: 1
Index: (6)
Kind: Ex

Name: The field generated by \(L\) over \(F\)
Definition: The smallest subfield of \(K\) containing both \(F\) and \(L\).
Page: 517
Chapter: 13
Section: 1
Variables: \(F\) a field,
\(K\) an extension of \(F\),
\(L\subseteq K\)
Notation: \(F(L)\)

Name: A simple extension of \(F\)
Definition: An extension \(K\) such that \(K=F(k)\) for some \(k\in K\).
Page: 517
Chapter: 13
Section: 1
Variables: \(F\) a field,

Name: A primitive element for the expansion
Definition: An element \(k\in K\) such that \(K=F(k)\).
Page: 517
Chapter: 13
Section: 1
Variables: \(F\) a field,
\(K\) a simple extension of \(F\)

Smaller0: \(p(k)=0\)
Greater0: \(F[x]/(p(x))\)
Page: 517
Section: 1
Variables: \(F\) a field,
\(K\) an extension of \(F\),
\(p(x)\in F[x]\),
\(k\in K\)
Index: 6
Kind: Theo
Assuming: \(p\) irreduccible
Greater1: is isomorphic to
Greater2: \(F(k)\)

Smaller0: there is an isomorphism \(\phi:F_0\to F_1\), sending \(\alpha_0\) to \(\alpha_1\), and \(\phi(p_0)=p_1\)
Greater0: \(F_0(\alpha_0)\) is isomorphic to \(F_1(\alpha_1)\)
Page: 519
Section: 1
Variables: \(F_0,F_1\) fields,
\(p_i(x)\in F_i[x]\)
\(K_i\) an extension of \(F_i\) in which \(p_i\) has a root \(\alpha_i\),
Index: 8
Kind: Theo
Hide middle: yes
Smaller1: the \(p_i\)'s are irreduccible

Name: An algebraic element of \(K\) over \(F\)
Definition: A root in \(K\) of some nonzero polynomial \(f(x)\in F[x]\)
Page: 520
Chapter: 13
Section: 2
Variables: \(F\) a field
\(K\) an extension of \(F\)

Name: A transcendental element of \(K\) over \(F\)
Definition: A non-algebraic element of \(K\) over \(F\)
Page: 520
Chapter: 13
Section: 2
Variables: \(F\) a field
\(K\) an extension of \(F\)

Name: An algebraic extension
Definition: An extension \(K/F\) whose every elements are algebraic over \(F\)
Page: 520, 527
Chapter: 13
Section: 2
Variables: \(F\) a field
Kind: Def, Theo
Typ: Transitive relation

Question: The number of monic irreducible polynomial in \(F[x]\) which has \(\alpha\) as a root
Answer: is exactly one
Page: 520
Section: 2
Variables: \(F\) a field,
\(K\) an extension
\(\alpha\in K\) algebraic over \(F\)
Index: 9
Kind: Prop

Name: The minimal polynomial for \(\alpha\) over \(F\)
Definition: The only monic irreducible polynomial in \(F[x]\) which has \(\alpha\) as a root
Page: 520
Chapter: 13
Section: 2
Variables: \(F\) a field,
\(K\) an extension
\(\alpha\in K\) algebraic over \(F\)
Notation: \(m_{\alpha,F}(x)\)
Index: 9
Kind: Prop, def
Notation2: \(m_{\alpha}(x)\) when \(F\) is clear

Smaller0: \(f\) has \(\alpha\) has a root
Greater0: divides \(f\)
Page: 520
Section: 2
Variables: \(F\) a field,
\(K\) an extension
\(\alpha\in K\) algebraic over \(F\),
\(f\in F[x]\)
Index: 9
Kind: Prop
Smaller to greater: iff
Prefix Greater: \(m_{\alpha,F}(x)\)

Smaller0: \(\alpha\) algebraic over both \(F\) and \(L\)
Greater0: \(m_{\alpha,L}(x)| m_{\alpha,F}(x)\)
Page: 520
Section: 2
Variables: \(F\) a field,
\(L\) an extension of \(F\),
\(K\) an extension of \(L\)
\(\alpha\in K\)
Index: 10
Kind: Cor
Counterexample: Assumes that \(\alpha\) is algebraic, but in an extension which does not contain \(L\)

Name: The degree of \(\alpha\)
Definition: The degree of \(m_{\alpha,F}(x)\)
Page: 520
Chapter: 13
Section: 2
Variables: \(F\) a field,
\(K\) an extension of \(F\)
\(\alpha\in K\) algebraic over \(F\)
Kind: Def

Smaller0: \(F(\alpha)\)
Greater0: \(F[x]/(m_\alpha(x))\)
Page: 521
Section: 2
Variables: \(F\) a field,
\(K\) an extension of \(F\)
\(\alpha\in K\) algebraic over \(F\)
Index: 11
Kind: Prop
Smaller to greater: is isomorphic to

Question: \([F(\alpha):F]\) equals
Answer: the degree of \(\alpha\)
Page: 521
Section: 2
Variables: \(F\) a field,
\(K\) an extension of \(F\)
\(\alpha\in K\) algebraic over \(F\)
Index: 11
Kind: Prop

Question: \(m_{\sqrt[n]2,\mathbb Q}(x)\)
Answer: equals \(x^n-2\)
Page: 521
Section: 2
Variables: \(n>1\)
Index: 3
Kind: Example

Question: The degree of \(\sqrt[n]2\) over \(\mathbb Q\) is
Answer: \(n\)
Page: 521
Section: 2
Variables: \(n>1\)
Index: 3
Kind: Example

Question: The degree of \(\sqrt[n]2\) over \(\mathbb R\) is
Answer: \(1\)
Page: 521
Section: 2
Variables: \(n>1\)
Index: 4
Kind: Example

Smaller0: algebraic over \(F\)
Greater0: finite
Page: 521
Section: 2
Variables: \(F\) a field,
\(K\) an extension of \(F\)
\(\alpha\in K\)
Index: 12
Kind: Prop
Smaller to greater: iff
Prefix Greater: the simple extension \(F(\alpha)/F\) is
Prefix Smaller: \(\alpha\) is

Smaller0: \(\alpha\) is an element of an extension of degree \(n\) over \(F\)
Greater0: \(\alpha\) satisfies a polynomial of degree at most \(n\) over \(F\)
Page: 521
Section: 2
Variables: \(F\) a field,
\(K\) an extension of \(F\)
\(\alpha\in K\)
Index: 12
Kind: Prop

Smaller0: \(\alpha\) satisfies a polynomial of degree \(n\)
Greater0: at most \(n\)
Page: 521
Section: 2
Variables: \(F\) a field,
\(K\) an extension of \(F\)
\(\alpha\in K\)
Index: 12
Kind: Prop
Prefix Greater: the degree of \(F(\alpha)\) over \(F\) is

Smaller0: \(K/F\) is finite
Greater0: \(K/F\) is algebraic
Page: 522
Section: 2
Variables: \(F\) a field,
\(K\) an extension of \(F\)
Kind: Cor
Extra: ex from wp
Counterexample: field of algebraic numbers

Smaller0: \(F\)'s characteristic is not 2
Greater0: \(K\)
Page: 522
Section: 2
Variables: \(F\) a field,
\(K\) an extension of \(F\),
\(\alpha\in K\setminus F\)
Kind: Ex
Smaller1: \(K\) is an extension of degree 2
Greater1: \(F(\alpha)\)
Connect Greater0: is isomorphic to

Name: A quadratic extension of \(F\)
Definition: An extension of degree \(2\)
Page: 522
Chapter: 13
Section: 2
Variables: \(F\) a field,
Kind: Ex
Assuming: \(F\)'s characteristic is not 2

Smaller0: \([L:F]\)
Greater0: \([L:K][K:F]\)
Page: 523
Section: 2
Variables: \(F\subseteq K\subseteq L\) fields
Index: 14
Kind: Theo
Smaller to greater: is equal to

Smaller0: \([K:F]\)
Greater0: \([L:F]\)
Page: 524
Section: 2
Variables: \(F\subseteq K\subseteq L\) fields
Index: 15
Kind: cor
Smaller to greater: divides
Assuming: \(L/F\) is a finite extension
Greater to smaller: is a multiple of

Name: A finitely generated extension
Definition: An extension of the form \(F(\alpha_1,\dots,\alpha_k)\) for some \(\alpha_i\) in the extension
Page: 524
Chapter: 13
Section: 2
Variables: \(F\) a field
Index: Def

Smaller0: \(F(\alpha,\beta)\)
Greater0: \((F(\alpha))(\beta)\)
Page: 525
Section: 2
Variables: \(F\subseteq K\) fields,
\(\alpha,\beta\in K\setminus F\)
Index: 16
Kind: Lem
Smaller to greater: =

Smaller0: \(x\in F(\alpha)\)
Greater0: \(\{\alpha^i\mid 0\le i<\text{deg}_F\alpha\}\)
Page: 526
Section: 2
Variables: \(F\subseteq K\) fields,
\(\alpha\in K\) over \(F\)
Smaller to greater: iff
Prefix Greater: \(x\) is a linear combination of
Assuming: \(\alpha\) algebraic

Question: \(x\) is a linear combination of
Answer: \(\{\alpha^i\mid i\in\mathbb Z\}\)
Page: 526
Section: 2
Variables: \(F\subseteq K\) fields,
\(\alpha\in K\) over \(F\)
\(x\in F(\alpha)\)
Index: Def

Smaller0: \(K/F\) is finite
Greater0: \(K\) is finitely generated by algebraic elements
Page: 526
Section: 2
Variables: \(F\subseteq K\) fields,
Index: 17
Kind: Theo
Smaller to greater: iff

Question: \(K/F\)'s degree is
Answer: at most \(\Pi_i\text{deg}(\alpha_i)\)
Page: 526
Section: 2
Variables: \(F\subseteq K\) fields,
\(\alpha_1,\dots,\alpha_n\) a set of generators of \(K\)
Index: 17
Kind: Theo

Name: The set of algebraic elements
Page: 527
Section: 2
Chapter: 13
Index: 18
Variables: \(F\) a field
Kind: Cor
Under: arithmetic operations, when they are defined

Name: The algebraic elements of \(L\) over \(F\)
Page: 57
Chapter: 13
Section: 2
Variables: \(F\subseteq L\) fields
Index: 19
Kind: Cor
Typ: A subfield of \(L\)

Name: The field of algebraic numbers
Page: 527
Chapter: 13
Section: 2
Notation: \(\overline{\mathbb Q}\)
Index: 1
Kind: Ex
Cardinal: Countable

Name: An algebraic number
Definition: A complex number which is algebraic over \(\mathbb Q\)
Page: 527
Chapter: 13
Section: 2
Index: 1
Kind: Ex

Name: The composite field of \(K_1\) and \(K_2\)
Definition: The smallest subfield of \(K\) containing \(K_1\) and \(K_2\)
Page: 527
Chapter: 13
Section: 2
Variables: \(K_i\subseteq K\) fields
Notation: \(K_1K_2\)
Kind: Def

Smaller0: \([K_1K_2:F]\)
Greater0: \([K_1:F][K_2:F]\)
Page: 528
Section: 2
Variables: \(K_i\subseteq F\) fields
Index: 21
Kind: Prop
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: a basis of a \(K_i\) over \(F\) remains linearly independant over \(K_{3-i}\)
Greater0: \([K_1K_2:F]=[K_1:F][K_2:F]\)
Page: 528
Section: 2
Variables: \(K_1,K_2\subseteq K\) fields
Index: 21, 22
Kind: Prop, Cor
Smaller1: the index of the \(K_i\)'s are relatively prime
Connect Smaller0: or

Name: An infinite extension
Definition: An extension whose degree is infinite
Page: 512
Chapter: 13
Section: 1
Variables: \(F\) a field

Question: Its prime subfield is
Answer: \(\mathbb Q\) if \(\text{ch}(F)=0\) and \(\mathbb F_{\text{ch}(F)}\) otherwise.
Page: 511
Section: 1
Variables: \(F\) a field
Kind: Def

Name: The prime subfield of \(F\)
Definition: The subfield of \(F\) generated by \(1_F\).
Page: 511
Chapter: 13
Section: 1
Variables: \(F\) a field
Kind: Def, ex
Example: \(\mathbb Q\)
Example2: \(\mathbb Q\).
Example3: \(\mathbb F_p\) as the set of constant polynomials.
Applied to3: (\mathbb F_p(x)\)
Applied to2: (\mathbb R\)

Name: The straightedges constructions
Definition: Connecting two given points by a straight line
Page: 532
Chapter: 13
Section: 3
Definition2: Finding a point of intersection of two straight lines
Conjdef: All of

Name: The compass constructions
Definition: drawing a circle with given radius and center
Page: 532
Chapter: 13
Section: 3
Definition2: finding the point(s) of intersection of a straight line and a circle or the intersection of two circles.
Conjdef: All of

Page: 532
Section: 3
Chapter: 13
Effect: Constructing a segment of length \(ab\)
Variables: \(a,b\) length

Page: 532
Section: 3
Chapter: 13
Effect: Constructing a segment of length \(a/b\)
Variables: \(a,b\) length

Page: 532
Section: 3
Chapter: 13
Effect: Constructing a segment of length \(\sqrt a\)
Variables: \(a\) length

Text: are:
-
Page: 532, 533
Chapter: 13
Section: 3
Name: Constructible lengths
Q: [...].
A: the four arithmetic operations, assuming it is defined and non-negative
Q: [...]
A: square root
Q: [...]
A: closed
Q: [...]
A: closed
Q: [...]
A: any operation not expressible by the above ones
Not3: not

Smaller0: \(\alpha\) is constructible by \(k\) straightedges and compass operations
Greater0: \([F(\alpha):F]\)
Page: 532
Section: 3
Variables: \(F\subseteq\mathbb R\),
\(k\in\mathbb N\),
\(\alpha\in\mathbb R\)
Index: 23
Kind: Prop
Greater1: a power of 2, by an integer at most \(k\)
Connect Greater0: is
Extra: counter example?

Question: Doubling the cube using ruler and compass is
Answer: impossible
Page: 532
Section: 3
Index: 24
Kind: Theo
Construction: It amounts to constructing \(\sqrt[3]2\) starting with 1. \([\mathbb Q(\sqrt[3]2):\mathbb Q]=3\) is not a power of \(2\)

Question: trisecting the angle using ruler and compass is
Answer: impossible
Page: 532
Section: 3
Index: 24
Kind: Theo
Construction: It allows to compute cosinus of constructible angle. And some angles \(\theta\) are contructible, such that \([\mathbb Q(\cos\theta):\mathbb Q]=3\)

Question: Squaring the Circle using ruler and compass is
Answer: impossible
Page: 532
Section: 3
Index: 24
Kind: Theo
Construction: \(\pi\)'s degree is infinite

Smaller0: \(\theta\) is constructible
Greater0: \(\theta\) is a multiple of 3
Page: 534
Section: 3
Variables: \(\theta\in\mathbb N\) an angle, in degree
Smaller to greater: iff
Context: Angles

Question: Why does ruler allows more expressivity than straightedge and compass ?
Answer: Putting one side of a ruler on a figure, some part of the ruler on another figure, and allowing them to slide along the first figure, one can use the ruler to find a line of distance 1 between the two figures. It is not doable with straightedge and compass.
Page: 535
Section: 3

Name: A splitting field for \(f\)
Definition: An extension \(K\) of \(F\), minimal such that \(f(x)\) splits completely in \(K[x]\)
Page: 536, , 541, 542
Chapter: 13
Section: 4
Variables: \(F\) a field
\(f\in F[x]\)
Index: , 25, 27, 28
Kind: Def, Theo, Theo, Cor,
Cardinal: Unique, up to iso

Name: A linear factor of \(f\)
Definition: A term of the form \(x-\alpha\) which divides \(f\)
Page: 536
Chapter: 13
Section: 4
Variables: \(F\) a field,
\(f\in F[x]\)

Name: A normal extension of \(F\)
Definition: The splitting field over \(F\) for a collection of polynomials in \(F[x]\)
Page: 537
Chapter: 13
Section: 4
Variables: \(F\) a field
Kind: Def

Question: The splitting field for \(x^2-2\) over \(\mathbb Q\) is
Answer: \(\mathbb Q(\sqrt 2)\)
Page: 537
Section: 4
Index: 1
Kind: Ex
Construction: there are only two roots, \(\pm\sqrt2\)

Question: The splitting field for \((x^2-2)(x^2-3)\) over \(\mathbb Q\) is
Answer: \(\mathbb Q(\sqrt 2,\sqrt 3)\)
Page: 537
Section: 4
Index: 2
Kind: Ex
Construction: \(\sqrt2\) and \(\sqrt3\) are roots.

Question: The splitting field for \((x^3-2)\) over \(\mathbb Q\) is
Answer: \(\mathbb Q(\sqrt[3] 2,\sqrt{-3})\)
Page: 537
Section: 4
Index: 3
Kind: Ex
Construction: Roots are:
* \(\sqrt[3]{2}\)
* \(\sqrt[3]{2}\left(\frac{-1+i\sqrt3}{2}\right)\)
* \(\sqrt[3]{2}\left(\frac{-1+-\sqrt3}{2}\right)\)

Question: The splitting field for \((x^4+4)\) over \(\mathbb Q\) is
Answer: \(\mathbb Q(i)\)
Page: 538
Section: 4
Kind: Ex
Construction: Root are \(\pm1\pm i\)

Smaller0: \([F[\alpha]:F]=\text{deg }f\)
Greater0: \(f(x)\) is irreduccible
Page: 538
Section: 4
Variables: \(F\) a field
\(K\) an extension
\(f\in F[x]\)
\(\alpha\) a root of \(f\)
Smaller to greater: iff

Question: \([F[\alpha]:F]\)
Answer: is at most \(n\)
Page: 538
Section: 4
Variables: \(F\) a field,
\(K\) an extension
\(f\in F[x]\) of degree \(n\)
\(\alpha\) a root of \(f\)

Question: The degree of the splitting field of \(f\) over \(F\) is
Answer: at most \(n!\)
Page: 538
Section: 4
Variables: \(F\) a field,
\(f\in F[x]\) of degree \(n\)
Index: 26
Kind: Prop

Name: The \(n\)^th_ roots of unity
Definition: The roots of the polynomial \(x^n-1\)
Page: 539
Chapter: 13
Section: 4
Variables: \(n\in\mathbb N\)
Kind: Ex
Definition2: \(e^{\frac{2\pi k i}n}\)

Question: The splitting field of \(x^n-1\) over \(\mathbb Q\) is viewed as
Answer: a subset of \(\mathbb C\) generated by \(e^{\frac{2\pi k i}n}\)
Page: 539
Section: 4
Variables: \(n\in\mathbb N\)
Kind: Ex

Name: A primitive \(n\)^th root of unity
Definition: A \(n\)^th root of unity which generate the cyclic group of all \(n\)^th roots of unity
Page: 539
Chapter: 13
Section: 4
Variables: \(n\in\mathbb N\)
Kind: Def
Denoted by: \(\xi_n\)

Name: The primitive \(n\)th root of unity
Page: 539
Section: 4
Chapter: 13
Variables: \(n\in\mathbb N\)
Kind: Def
Under: power by an integer coprime with \(n\)

Question: The number of primitive \(n\)^th root of unity
Answer: is \(\phi(n)\) with \(\phi\) being Euler's function
Page: 539
Section: 4
Variables: \(n\in\mathbb N\)
Kind: Def

Name: The cyclotomic field of \(n\)^th roots of unity
Definition: \(\mathbb Q(\xi_n)\)
Page: 540
Chapter: 13
Section: 4
Variables: \(n\in\mathbb N\)
Kind: Def

Question: \([\mathbb Q(\xi_n):\mathbb Q]\)
Answer: equals \(\phi(n)\)
Page: 540
Section: 4
Variables: \(n\in\mathbb N\)

Name: The algebraic closure of \(F\)
Definition: An algebraic extension \(G\) such that every polynomials over \(F\) splits completely over \(G\)
Page: 543, 544
Chapter: 13
Section: 4
Variables: \(F\) a field
Notation: \(\overline F\)
Index: , 29, 31
Kind: Def, Prop, Prop
Typ: An algebraically closed field
Cardinal: Unique up to iso

Name: An algebraically closed field
Definition: every polynomial has a root in this field.
Page: 543, 545
Chapter: 13
Section: 4
Kind: Def, Theo
Definition2: every polynomial splits completely over it
Example: \(\mathbb C\)
Conjdef: A field in which, equivalently

Question: How to construct the algebraic closure of \(F\) ?
Answer: Takes algebraic elements of \(\bigcup_{j\ge 0}K_j\), with:
-\(K_0=F\)
-\(K_{j+1}=K_{j}[(x_f)_{f\in\mathcal F_j}]/\mathcal M_{j}\), with
--\(\mathcal F_j\) being the set of polynomials over \(K_j\),
--\(\mathcal M_j\) the maximal ideal containing \(f(x_f)\) for each \(f\in K_j\)
Page: 544
Section: 4
Variables: \(F\) a field
Index: 30, 31
Kind: Prop

Name: A simple root of \(f\)
Definition: An element \(\alpha_i\) such that \(n_i=1\) when an element \(\alpha_i\) such that \(n_i>1\), when
\(K\) a splitting field for \(f\)
\(f(x)=\prod(x-\alpha_i)^{n_i}\)
Page: 545
Chapter: 13
Section: 5
Variables: \(F\) a ring
\(f\in F[x]\)
Kind: Theo

Name: A multiple root of \(f\)
Definition: \(n_i>1\), when
\(K\) a splitting field for \(f\)
\(f(x)=\prod(x-\alpha_i)^{n_i}\)
Page: 545, 547
Chapter: 13
Section: 5
Variables: \(F\) a ring
\(f\in F[x]\)
Index: , 33
Kind: , Prop
Definition2: is also a root of \(D_xf(x)\)
Conjdef: An element \(\alpha_i\) such that, equivalently

Name: A separable polynomial over \(F\)
Definition: without multiple root
Page: 545, 547, 549, 550
Chapter: 13
Section: 5
Variables: \(F\) a ring
Index: , (1), (2), 34, 37
Kind: , Ex, cor, Prop
Definition2: a product of distinct irreducible polynomials (assuming perfect field)
Example: \(x^2-2\) over \(\mathbb Q\)
Conjdef: equivalently
Definition3: whose inseparable degree is \(1\)
Example2: \(x^{p^n}-x\) over \(\mathbb F^p\)
Definition4: whose degree equals its separable degree
Example3: \(x^n-1\), assuming the field's characteristic does not divide \(n\)
DefType: A polynomial

Name: An inseparable polynomial over \(F\)
Definition: have multiple root
Page: 545
Chapter: 13
Section: 5
Variables: \(F\) a ring
Kind: Theo
Definition2: is not separable
Example: \((x^2-2)^n\) over \(\mathbb Q\)
Conjdef: equivalently
Example2: \(x^2-t\) over the field \(\mathbb F_2(t)\) of rational functions in \(t\)
DefType: A polynomial

Name: The derivative of \(f\)
Definition: \(\sum_{i=0}^{n-1}(i+1)a_{i+1}x^i\)
Page: 546
Chapter: 13
Section: 5
Variables: \(F\) a ring
\(f(x)=\sum_{i=0}^n{a_i}x^{i}\)
Notation: \(D_xf(x)\)
Kind: Theo

Question: \(D_x(\sum\lambda_jf_j(x))\)
Answer: \(\sum\lambda_jD_xf_j(x)\)
Page: 546
Section: 5
Variables: \(F\) a ring
\(f_j(x)=\sum_{i=0}^{n_j}{a_{i,j}}x^{i}\)
\(\lambda_j\in F\)
Kind: Theo

Question: \(D_x(\prod f_j(x))\)
Answer: \(\sum_j D_xf_j(x)\prod_{k\ne j} f_{k}(x)\)
Page: 546
Section: 5
Variables: \(F\) a ring
\(f_j(x)=\sum_{i=0}^{n_j}{a_{i,j}}x^{i}\)
Context: Derivative in \(x\)

Question: \(D_x\left(x^{p^n}-x\right)\) equals
Answer: \(p^nx^{p^n-1}-1=-1\)
Page: 547
Section: 5
Variables: \(p\) prime
Context: \(\mathbb Z/p\mathbb Z\)
Index: (1)
Kind: Ex

Smaller0: \(D_xf=0\)
Greater0: \(\sum a_ix^{ip}\)
Page: 547, 548
Section: 5
Variables: \(F\) a field of characteristic \(p\)
\(f\in F[x]\)
Smaller to greater: iff
Prefix Greater: \(f\) is of the form

Smaller0: \((a+b)^p\)
Greater0: \(a^p+b^p\)
Page: 548
Section: 5
Variables: \(F\) a field of characteristic \(p\)
\(a,b\in F\)
Index: 35
Kind: Prop
Smaller to greater: =
Context: \(\mathbb Z/p\mathbb Z\)

Smaller0: \((ab)^p\)
Greater0: \(a^pb^p\)
Page: 548
Section: 5
Variables: \(F\) a field of characteristic \(p\)
\(a,b\in F\)
Index: 35
Kind: Prop
Smaller to greater: =
Context: \(\mathbb Z/p\mathbb Z\)

Name: The Frobenius endomorphism of \(F\)
Definition: [$]\defFun{F}{F}{a}{a^p}[/$] for \(p\) the characteristic of the field
Page: 549
Chapter: 13
Section: 5
Variables: \(F\) a field
Kind: Def

Smaller0: \(F\) is finite
Greater0: an isomorphism
Page: 549
Section: 5
Variables: \(F\) a field of characteristic \(p\)
Index: 36
Kind: Cor
Prefix Greater: the Frobenius endomorphism of \(F\) is

Name: A perfect field
Definition: its characteristic is \(0\)
Page: 549, https://en.wikipedia.org/wiki/Perfect_field
Chapter: 13
Section: 5
Index: 36
Kind: def, Cor
Definition2: its characteristic is \(p\) and \(\{x^p\mid x\in F\}=F\)
Example: Algebraically closed fields
Conjdef: A field, such that either
Example2: \(\mathbb Q\), \(\mathbb R\), \(\mathbb C\)
Example3: Finite fields
Counterexample: \(\mathbb F_q(X)\)

Definition: The splitting field of \(x^{p^n}-x\) over \(\mathbb F_p\)
Page: 549
Chapter: 13
Section: 5
Notation: \(\mathbb F_{p^n}\)
Kind: Ex
Definition2: The set of roots of \(x^{p^n}-x\) in an extension of \(\mathbb F_p\)
Definition3: The only field of order \(p^n\)
Assuming: \(p\) prime

Smaller0: there is a field of order \(n\)
Greater0: \(n\) is a power of a prime
Page: 549
Section: 5
Variables: \(n\in\mathbb N\)
Kind: Ex
Smaller to greater: iff
Intermediate: there is exactly one field of order \(n\)

Question: The normal form of an irreduccible polynomial is
Answer: \(p_{\text{sep}}(x^{p^k})\) with \(p_{\text{sep}}\) irreducible separable polynomial, and \(k\in\mathbb N\)
Page: 550
Section: 5
Variables: \(F\) a field of characteristic \(p\)
Context: Characteristic \(p\)
Index: 38
Kind: Prop

Name: The separable degree of \(q(x)\)
Definition: The degree of \(p_{\text{sep}}\) such that \(q=p_{\text{sep}}(x^{p^k})\)
Page: 550
Chapter: 13
Section: 5
Variables: \(F\) a field of characteristic \(p\)
\(q\in F[x]\)
Notation: \(\text{deg}_s q(x)\)
Kind: Def
Assuming: \(q\) irreducible

Name: The inseparable degree of \(q(x)\)
Definition: \(p^k\) such that \(q=p_{\text{sep}}(x^{p^k})\)
Page: 550
Chapter: 13
Section: 5
Variables: \(F\) a field of characteristic \(p\)
\(q\in F[x]\)
Notation: \(\text{deg}_i q(x)\)
Kind: Def
Assuming: \(q\) irreducible

Question: What is the relation between \(\text{deg}_s q(x)\) and \(\text{deg}_i q(x)\)
Answer: \(\text{deg}_i q(x)\text{deg}_s q(x)=\text{deg}q(x)\)
Page: 550
Section: 5
Variables: \(F\) a field of characteristic \(p\)
\(q\in F[x]\)
Kind: Def
Assuming: \(q\) irreducible

Question: In \(\mathbb F_2(t)\), the normal form of \(x^{2^m}-t\) is
Answer: \(\left(x\mapsto (x-t)\right)(x^{2^m})\)
Page: 551
Section: 5
Index: 2
Kind: Ex

Name: The separable polynomial part of \(q\)
Definition: A polynomial \(r\) such that \(q(x)=r(x^{p^k})\) for some \(k\in\mathbb N\)
Page: 551
Chapter: 13
Section: 5
Variables: \(F\) a field of characteristic \(p\)
\(q\in F[x]\)
Notation: \(q_{sep}\)
Index: 2
Kind: Ex
Assuming: \(q\) irreducible

Smaller0: \(F\) perfect field
Greater0: \(K\) is separable
Page: 551
Section: 5
Variables: \(F\) a field
\(K\) an extension of \(F\)
Index: 39
Kind: Cor
Smaller1: \([K:F]\) is finite

Page: 513
Chapter: 13
Section: 1
Variables: \(F\) a field,
\(p[x]\in F[x]\), irreduccible
Notation: \(F[x]/(p(x))\)
Kind: Theo
Index: 4
Base: \(\{x^i\mid 0\le i<\text{deg} p-1\}\)
Field-base: \(F\)

Smaller0: \(\mathbb Q[x]/(x^2-2)\)
Greater0: \(\mathbb Q[\sqrt2]\)
Page: 513
Section: 1
Index: (3)
Kind: Ex
Smaller to greater: is isomorphic to

Page: 525
Chapter: 13
Section: 2
Variables: \(F\subseteq K\) fields,
\(\alpha,\beta\in K\) algebraic over \(F\)
Notation: \(F(\alpha,\beta)\)
Kind: Lem
Index: 16
Base: \(\{\alpha^i\beta^j\mid 0\le i<\text{deg}_F(\alpha),0\le j<\text{deg}_{F(\alpha)}(\beta)\}\)
Field-base: \(F\)