nyuuzyou/wikis huggingface dataset:
Prime to Own Power minus 1 over Prime minus 1 being Prime

== Theorem ==
Let $n \in \Z_{>1}$ be an integer greater than $1$.
Then $\dfrac {n^n - 1} {n - 1}$ is a prime for $n$ equal to:
:$2, 3, 19, 31$
== Proof ==
Note that if $4 p + 1$ is prime for prime $p$, then $\dfrac {p^p - 1} {p - 1}$ is divisible by $4 p + 1$:
Let $q = 4 p + 1$ be prime.
By First Supplement to Law of Quadratic Reciprocity:
:$\paren {\dfrac {-1} q} = 1$
that is, there exists some integer $I$ such that $I^2 \equiv -1 \pmod q$.
Then:
and thus:
Hence $q \divides \paren {p^p - 1}$.
Obviously $q > p - 1$.
Therefore $q \divides \dfrac {p^p - 1} {p - 1}$.
== Sources ==
* : $\mathbf A$: Prime Numbers: $\mathbf {A 3}$: Mersenne primes. Repunits. Fermat numbers. Primes of shape $k \cdot 2^n + 1$.
* : $19$
* : $\mathbf A$: Prime Numbers: $\mathbf {A 3}$: Mersenne primes. Repunits. Fermat numbers. Primes of shape $k \cdot 2^n + 1$.
Category:Prime Numbers
Prime to Own Power minus 1 over Prime minus 1 being Prime/Mistake/Guy/Second Edition

== Source Work ==
:
:$\mathbf A$: Prime Numbers
::$\mathbf {A 3}$: Mersenne primes. Repunits. Fermat primes. Primes of shape $k \cdot 2^n + 1$.
== Mistake ==
: observes that the only primes $< 180$ for which $\paren {p^p - 1} / \paren {p - 1}$ is prime are $2$, $3$, $7$, $19$ and $31$; ...
== Correction ==
Untrue:
:$\dfrac {7^7 - 1} {7 - 1} = 137257 = 29 \times 4733$
This has been corrected in the $3$rd edition.
== Sources ==
* : $\mathbf A$: Prime Numbers: $\mathbf {A 3}$: Mersenne primes. Repunits. Fermat numbers. Primes of shape $k \cdot 2^n + 1$.
Category:Mistakes/Unsolved Problems in Number Theory
Prime to Own Power minus 1 over Prime minus 1 being Prime/Mistake/Wells

== Source Work ==
:
:The Dictionary
::$19$
== Mistake ==
:$\paren {19^{19} - 1} / \paren {19 - 1}$ is prime. Other primes with this property are $2$, $3$, $7$ and $31$.
Untrue:
:$\dfrac {7^7 - 1} {7 - 1} = 137257 = 29 \times 4733$
This mistake was inherited from : $\mathbf A 3$.
== Sources ==
* : $19$
Category:Mistakes/Curious and Interesting Numbers
Prime whose Divisor Sum is Square is 3

== Theorem ==
There is exactly $1$ prime number whose divisor sum is a square number, and that is $3$:
:$\map {\sigma_1} 3 = 4$
== Proof ==
That $\map {\sigma_1} 3 = 4$ is shown at .
It remains to be shown there are no more.
Let $n \in \N$ such that $\map {\sigma_1} n$ is square.
:$\map {\sigma_1} n = m^2$
Suppose $n$ is prime.
From Divisor Sum of Prime Number:
:$\map {\sigma_1} n = n + 1$
So we have:
But $\paren {m + 1} \paren {m - 1}$ is prime $m - 1 = 1$, that is, $m = 2$.
That is the case covered by $\map {\sigma_1} 3 = 4$.
Hence the result.
Category:Divisor Sum Function
Category:3
Primes Expressible as x^2 + n y^2 for all n from 1 to 10

== Sequence ==
The sequence of prime numbers which can be expressed in the form $x^2 + n y^2$ for all values of $n$ from $1$ to $10$ begins:
:$1009, 1129, 1201, 1801, \ldots$
This sequence appears not to be documented on .
== Proof ==
== Examples ==
== Historical Note ==
== Sources ==
* 
* : $1009$
Category:Prime Numbers
Category:Polynomial Expressions for Primes
Category:Primes Expressible as x^2 + n y^2 for all n from 1 to 10
Primes Expressible as x^2 + n y^2 for all n from 1 to 10/Examples

== Examples of Primes Expressible as $x^2 + n y^2$ for all $n$ from $1$ to $10$ ==
=== 1009 ===
=== 1129 ===
=== 1201 ===
=== 1801 ===
Category:Primes Expressible as x^2 + n y^2 for all n from 1 to 10
Primes Expressible as x^2 + n y^2 for all n from 1 to 10/Examples/1009

== Example of Prime Expressible as $x^2 + n y^2$ for all $n$ from $1$ to $10$ ==
Category:Primes Expressible as x^2 + n y^2 for all n from 1 to 10
Category:1009
Primes Expressible as x^2 + n y^2 for all n from 1 to 10/Examples/1129

== Example of Prime Expressible as $x^2 + n y^2$ for all $n$ from $1$ to $10$ ==
Category:Primes Expressible as x^2 + n y^2 for all n from 1 to 10
Category:1129
Primes Expressible as x^2 + n y^2 for all n from 1 to 10/Examples/1201

== Example of Prime Expressible as $x^2 + n y^2$ for all $n$ from $1$ to $10$ ==
Category:Primes Expressible as x^2 + n y^2 for all n from 1 to 10
Category:1201
Primes Expressible as x^2 + n y^2 for all n from 1 to 10/Examples/1801

== Example of Prime Expressible as $x^2 + n y^2$ for all $n$ from $1$ to $10$ ==
Category:Primes Expressible as x^2 + n y^2 for all n from 1 to 10
Category:1801
Primes Expressible as x^2 + n y^2 for all n from 1 to 10/Historical Note

== Historical Note on Primes Expressible as $x^2 + n y^2$ for all $n$ from $1$ to $10$ ==
 conjectured that:
:The number $1201$ seems to be the smallest prime which can be expressed in the form $x^2 + n y^2$ for all values of $n$ from $1$ to $10$.
 reported in his of $1986$ that this hypothesis was presented in Volume $8$ of , but research is needed to track down the exact issue, date and page number.
In $1992$, reported in that the $1201$ is not in fact the smallest, but that $1009$ and $1129$ also have this property.
In his of $1997$, provided the above update. 
== Sources ==
* : $1201$
* : $1009$
Category:Historical Notes
Primes for which Powers to Themselves minus 1 have Common Factors

== Theorem ==
Let $p$ and $q$ be prime numbers such that $p^p - 1$ and $q^q - 1$ have a common divisor $d$.
The only known $p$ and $q$ such that $d < 400 \, 000$ are $p = 17, q = 3313$.
== Proof ==
This is of course rubbish, because $p^p - 1$ and $q^q - 1$ have the obvious common factor $2$.
Investigation ongoing, as it appears the author of the work this was plundered from must have been thinking about the Feit-Thompson Conjecture and got it badly wrong.
== Sources ==
* 
* : $17$
Category:Prime Numbers
Primes not Sum of or Difference between Powers of 2 and 3

== Theorem ==
The sequence of prime numbers which cannot be expressed as either sum of or the difference between a power of $2$ and a power of $3$ begins:
:$53, 71, 103, 107, 109, 149, 151, \ldots$
== Proof ==
== Sources ==
* : $53$
Category:Prime Numbers
Primes not Sum of or Difference between Powers of 2 and 3/Mistake

== Source Work ==
:
:The Dictionary
::$53$
== Mistake ==
:[$53$ is] The smallest prime which is not the difference between powers of $2$ and $3$. The sequence of such primes continues: $71 \quad 103 \quad 107 \quad 109 \quad 149 \quad 151 \ldots$
== Correction ==
This should read:
:[$53$ is] The smallest prime which is neither the sum of nor difference between powers of $2$ and $3$. The sequence of such primes continues: $71 \quad 103 \quad 107 \quad 109 \quad 149 \quad 151 \ldots$
We note that $43 = 2^5 + 3^2$ but cannot be expressed as the difference between powers of $2$ and $3$.
== Sources ==
* : $53$
Category:Mistakes/Curious and Interesting Numbers
Primes of Form n^2 + 1

== Sequence ==
The sequence of prime numbers of the form $n^2 + 1$ begins:
:$2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, \ldots$
These correspond to values of $n$ beginning:
:$1, 2, 4, 6, 10, 14, 16, 20, 24, \ldots$
== Sources ==
* : $1,3727 \ldots$
* : $257$
Category:Prime Numbers
Primes of form Power Less One

== Theorem ==
Let $m, n \in \N_{>0}$ be natural numbers.
Let $m^n - 1$ be prime.
Then $m = 2$ and $n$ is prime.
== Proof ==
First we note that by Integer Less One divides Power Less One:
:$\paren {m - 1} \divides \paren {m^n - 1}$
where $\divides$ denotes divisibility.
Thus $m^n - 1$ is composite for all $m \in \Z: m > 2$.
Let $m = 2$, and consider $2^n - 1$.
Suppose $n$ is composite.
Then $n = r s$ where $r, s \in \Z_{> 1}$.
Then by the corollary to Integer Less One divides Power Less One:
:$\paren {2^r - 1} \divides \paren {2^{r s} - 1}$
Thus if $n$ is composite, then so is $2^n - 1$.
So $2^n - 1$ can be prime only when $n$ is prime.
== Also see ==
Primes of the form $2^n - 1$ are called Mersenne primes.
They are particularly interesting because there is a convenient algorithm (the Lucas-Lehmer Test) which can determine the primality of such a number with high computational efficiency.
Hence, at the present time, the largest primes known are Mersenne.
== Historical Note ==
== Sources ==
* : Book $\text{IX}$. Proposition $36$: Footnote
* : $\S 2.4$: The rational numbers and some finite fields: Further Exercises $7$
* : $127$
* : Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes
* : $\S 1.14$: Exercise $18 \ (5)$
* : $127$
* 
* 
* 
* 
Category:Mersenne Primes
Category:Prime Numbers
Category:Mersenne Numbers
Primes of form Power Less One/Historical Note

== Historical Note on Primes of form Power Less One ==
The proof that if $n$ is composite, then so is $2^n - 1$, is historically attributed to , who gave it in $1603$.
 was aware of this result, but took it further to ask the question as to which numbers of the form $2^p - 1$ are prime when $p$ is prime.
== Sources ==
* : Chapter $\text {A}.12$: Mersenne ()
* : $127$
* : $127$
Category:Historical Notes
Primes of form Power of Two plus One

== Theorem ==
Let $n \in \N$ be a natural number.
Let $2^n + 1$ be prime.
Then $n = 2^k$ for some natural number $k$.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Definition:Fermat Prime: a prime of the form $2^{\paren {2^k} } + 1$.
== Historical Note ==
Category:Prime Numbers
Category:Fermat Numbers
Category:Primes of form Power of Two plus One
Primes of form Power of Two plus One/Historical Note

== Historical Note on Primes of form Power of Two plus One ==
In $1640$, wrote to wrote to announcing this result.
From the fact that all integers of the form $2^n + 1$ such that $n = 2^k$ that he tested were prime, he went on to make his famous Fermat Prime Conjecture: that they are all prime.
This was refuted by , who discovered that $2^{2^5}$ is composite in $1732$.
== Sources ==
* : $257$
* : $257$
Category:Historical Notes
Primes of form Power of Two plus One/Proof 1

== Theorem ==
== Proof ==
Suppose $n$ has an odd divisor apart from $1$.
Then $n$ can be expressed as $n = \paren {2 r + 1} s$.
So:
and so $2^n + 1$ is not prime.
Hence $2^n + 1$ can be prime only if $n$ has only even divisors.
That is, if $n = 2^k$ for some natural number $k$.
Category:Primes of form Power of Two plus One
Primes of form Power of Two plus One/Proof 2

== Theorem ==
== Proof ==
A specific instance of Primes of form Power plus One:
$q^n + 1$ is prime only if:
:$(1): \quad q$ is even
and
:$(2): \quad n$ is of the form $2^k$ for some positive integer $k$.
As $2$ is even, the result applies.
Category:Primes of form Power of Two plus One
Primes of form Power plus One

== Theorem ==
Let $q, n \in \Z_{>0}$ such that $q > 1$.
Then $q^n + 1$ is prime only if:
:$(1): \quad q$ is even
and
:$(2): \quad n$ is of the form $2^m$ for some positive integer $m$.
== Proof ==
Note that if $q = 1$ then $q^n + 1 = 2$ which is prime.
Hence the condition on $q$ in the statement of the theorem.
So by hypothesis $q > 1$.
Let $q$ be odd.
Then by Two divides Power Plus One iff Odd, $q^n + 1$ is not prime.
Let $q$ be even.
Let $n$ be expressed in the form $r 2^m$ where $r$ is odd.
Then $q^n + 1$ can be expressed in the form:
: $q^{r 2^m} + 1 = \left({q^{2^m}}\right)^r + 1$
By Number Plus One divides Power Plus One iff Odd, $q^{2^m} + 1$ is a divisor of $\left({q^{2^m}}\right)^r + 1$.
So for all $r > 1$ it follows that $\left({q^{2^m}}\right)^r + 1$ is composite.
Hence the result.
== Examples ==
The sequence of primes of the form $q^2 + 1$ begins:
The sequence of primes of the form $q^4 + 1$ begins:
== Also see ==
* Primes of form Power of Two plus One
== Sources ==
* : $\S 2.4$: The rational numbers and some finite fields: Further Exercises $8$
Category:Number Theory
Primes whose Digits are Consecutive Ascending from 1

== Sequence ==
The prime numbers consisting of strings of consecutive ascending digits starting from $1$ (allowing either $0$ or $1$ to follow $9$) begins:
:$1 \, 234 \, 567 \, 891, 12 \, 345 \, 678 \, 901 \, 234 \, 567 \, 891, 1 \, 234 \, 567 \, 891 \, 234 \, 567 \, 891 \, 234 \, 567 \, 891$
It is not known whether there exist any more.
== Historical Note ==
== Sources ==
* : $1,234,567,891$
* : $1,234,567,891$
Category:Prime Numbers
Primes whose Digits are Consecutive Ascending from 1/Historical Note

== Historical Note on Primes whose Digits are Consecutive Ascending from 1 ==
According to in his $1986$ work , this set of $3$ primes was reported by in Volume $10$, but details are lacking.
== Sources ==
* : $1,234,567,891$
* : $1,234,567,891$
Category:Historical Notes
Primitive Abundant Number/Examples/104

== Example of Primitive Abundant Number ==
$104$ is a primitive abundant number:
:$1 + 2 + 4 + 8 + 13 + 26 + 52 = 106 > 104$
== Proof ==
From , we have:
:$\map {\sigma_1} {104} - 104 = 106$
where $\sigma_1$ denotes the divisor sum function.
Thus, by definition, $104$ is an abundant number.
The aliquot parts of $104$ are enumerated at :
:$1, 2, 4, 8, 13, 26, 52$
By inspecting the divisor sums of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:104
Primitive Abundant Number/Examples/20

== Example of Primitive Abundant Number ==
$20$ is a primitive abundant number:
:$1 + 2 + 4 + 5 + 10 = 22 > 20$
== Proof ==
From , we have:
:$\map {\sigma_1} {20} - 20 = 22$
where $\sigma_1$ denotes the divisor sum function.
Thus, by definition, $20$ is abundant number.
The aliquot parts of $20$ are enumerated at :
:$1, 2, 4, 5, 10$
By inspecting the divisor sum values of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:20
Primitive Abundant Number/Examples/272

== Example of Primitive Abundant Number ==
$272$ is a primitive abundant number:
:$1 + 2 + 4 + 8 + 16 + 17 + 34 + 68 + 136 = 286 > 272$
== Proof ==
From , we have:
:$\map {\sigma_1} {272} - 272 = 286$
where $\sigma_1$ denotes the divisor sum function: the sum of all divisors of $272$.
Thus, by definition, $272$ is an abundant number.
The aliquot parts of $272$ are enumerated at :
:$1, 2, 4, 8, 16, 17, 34, 68, 136$
By inspecting the divisor sum of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:272
Primitive Abundant Number/Examples/304

== Example of Primitive Abundant Number ==
$304$ is a primitive abundant number:
:$1 + 2 + 4 + 8 + 16 + 19 + 38 + 76 + 152 = 316 > 304$
== Proof ==
From , we have:
:$\map {\sigma_1} {304} - 304 = 316$
where $\sigma_1$ denotes the divisor sum function: the sum of all divisors of $304$.
Thus, by definition, $304$ is an abundant number.
The aliquot parts of $304$ are enumerated at :
:$1, 2, 4, 8, 16, 19, 38, 76, 152$
By inspecting the divisor sums of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:304
Primitive Abundant Number/Examples/368

== Example of Primitive Abundant Number ==
$368$ is a primitive abundant number:
:$1 + 2 + 4 + 8 + 16 + 23 + 46 + 92 + 184 = 376 > 368$
== Proof ==
From , we have:
:$\map {\sigma_1} {368} - 368 = 376$
where $\sigma_1$ denotes the divisor sum function.
Thus, by definition, $368$ is an abundant number.
The aliquot parts of $368$ are enumerated at :
:$1, 2, 4, 8, 16, 23, 46, 92, 184$
By inspecting the divisor sums of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:368
Primitive Abundant Number/Examples/464

== Example of Primitive Abundant Number ==
$464$ is a primitive abundant number:
:$1 + 2 + 4 + 8 + 16 + 29 + 58 + 116 + 232 = 466 > 464$
== Proof ==
From :
:$\map {\sigma_1} {464} - 464 = 466$
where $\sigma_1$ denotes the divisor sum function: the sum of all divisors of $464$.
Thus, by definition, $464$ is an abundant number.
The aliquot parts of $464$ are enumerated at :
:$1, 2, 4, 8, 16, 29, 58, 116, 232$
By inspecting the divisor sum of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:464
Primitive Abundant Number/Examples/550

== Example of Primitive Abundant Number ==
$550$ is a primitive abundant number:
:$1 + 2 + 5 + 10 + 11 + 22 + 25 + 50 + 55 + 110 + 275 = 566 > 550$
== Proof ==
From , we have:
:$\map {\sigma_1} {550} - 550 = 566$
where $\sigma_1$ denotes the divisor sum function,
Thus, by definition, $550$ is an abundant number.
The aliquot parts of $550$ are enumerated at :
:$1, 2, 5, 10, 11, 22, 25, 50, 55, 110, 275$
By inspecting the divisor sums of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:550
Primitive Abundant Number/Examples/572

== Example of Primitive Abundant Number ==
$572$ is a primitive abundant number:
:$1 + 2 + 4 + 11 + 13 + 22 + 26 + 44 + 52 + 143 + 286 = 604 > 572$
== Proof ==
From , we have:
:$\map {\sigma_1} {572} - 572 = 604$
where $\sigma_1$ denotes the divisor sum function.
Thus, by definition, $572$ is an abundant number.
The aliquot parts of $572$ are enumerated at :
:$1, 2, 4, 11, 13, 22, 26, 44, 52, 143, 286$
By inspecting the divisor sums of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:572
Primitive Abundant Number/Examples/650

== Example of Primitive Abundant Number ==
$650$ is a primitive abundant number:
:$1 + 2 + 5 + 10 + 13 + 25 + 26 + 50 + 65 + 130 + 325 = 652 > 650$
== Proof ==
From , we have:
:$\map {\sigma_1} {650} - 650 = 652$
where $\sigma_1$ denotes the divisor sum function: the sum of all divisors of $650$.
Thus, by definition, $650$ is an abundant number.
The aliquot parts of $650$ are enumerated at :
:$1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325$
By inspecting the divisor sum of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:650
Primitive Abundant Number/Examples/70

== Example of Primitive Abundant Number ==
$70$ is a primitive abundant number:
:$1 + 2 + 5 + 7 + 10 + 14 + 35 = 74 > 70$
== Proof ==
From , we have:
:$\map {\sigma_1} {70} - 70 = 74$
where $\sigma_1$ denotes the divisor sum.
Thus, by definition, $70$ is an abundant number.
The aliquot parts of $70$ are enumerated at :
:$1, 2, 5, 7, 10, 14, 35$
By inspecting the divisor sum of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:70
Primitive Abundant Number/Examples/748

== Example of Primitive Abundant Number ==
$748$ is a primitive abundant number:
:$1 + 2 + 4 + 11 + 17 + 22 + 34 + 44 + 68 + 187 + 374 = 764 > 748$
== Proof ==
From , we have:
:$\map {\sigma_1} {748} - 748 = 764$
where $\sigma_1$ denotes the divisor sum function.
Thus, by definition, $748$ is abundant number.
The aliquot parts of $748$ are enumerated at :
:$1, 2, 4, 11, 17, 22, 34, 44, 68, 187, 374$
By inspecting the divisor sums of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:748
Primitive Abundant Number/Examples/836

== Example of Primitive Abundant Number ==
$836$ is a primitive abundant number:
:$1 + 2 + 4 + 11 + 19 + 22 + 38 + 44 + 76 + 209 + 418 = 844 > 836$
== Proof ==
From , we have:
:$\map {\sigma_1} {836} - 836 = 844$
where $\sigma_1$ denotes the divisor sum function: the sum of all divisors of $836$.
Thus, by definition, $836$ is abundant number.
The aliquot parts of $836$ are enumerated at :
:$1, 2, 4, 11, 19, 22, 38, 44, 76, 209, 418$
By inspecting the divisor sums of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:836
Primitive Abundant Number/Examples/88

== Example of Primitive Abundant Number ==
$88$ is a primitive abundant number:
:$1 + 2 + 4 + 8 + 11 + 22 + 44 = 92 > 88$
== Proof ==
From , we have:
:$\map {\sigma_1} {88} - 88 = 92$
where $\sigma_1$ denotes the divisor sum function.
Thus, by definition, $88$ is an abundant number.
The aliquot parts of $88$ are enumerated at :
:$1, 2, 4, 8, 11, 22, 44$
By inspecting the divisor sums of each of these, they are seen to be deficient.
Hence the result, by definition of primitive abundant number.
Category:Primitive Abundant Numbers/Examples
Category:88
Primitive Complex Roots of Unity/Examples/Cube Roots

== Examples of Primitive Complex Roots of Unity ==
The primitive complex cube roots of unity are:
== Proof ==
There are $3$ (complex) cube roots of unity:
:$1, \omega, \omega^2$
We have:
Also:
Trivially, $1$ is not a primitive complex cube root of unity because you cannot make either $\omega$ or $\omega^2$ by multiplying $1$ by itself as many times as you like.
Hence the result.
== Sources ==
* 
* 
Category:Examples of Primitive Complex Roots of Unity
Category:Cube Roots of Unity
Primitive Element Theorem

== Theorem ==
Let $E / F$ be a separable field extension of finite degree.
Then $E / F$ is simple: there exists $\alpha\in E$ such that $E = \map F \alpha$.
== Proof ==
If $F$ is a finite field (equivalently $E$ is a finite field), this follows from Finite Extension of $\F_p$ is Generated By a Single Element, since the generator of $E / \F_p$ also generates $E / F$.
Next, assume $F$ is infinite.
Choose an algebraic closure $\overline F$ of $F$.
Let $\sigma_1, \dots, \sigma_n$ be all distinct embeddings of $E$ into $\overline F$ fixing $F$.
By definition 2 of separable degree,
:$n = \index E F_s$
Since $E / F$ is separable,
:$n = \index E F_s = \index E F$
Note that if $i \ne j$, then $\sigma _ i \ne \sigma _ j$, so
:$V_{ij} = \map \ker {\sigma _ i - \sigma _ j}$
is not equal to $E$.
By Vector Space over an Infinite Field is not equal to the Union of Proper Subspaces,
:$\ds \bigcup_{1\le i < j\le n} V_{ij}$
is a proper subfield of $E$,
so we can find $\alpha \in E$ with $\alpha \notin V_{ij}$ for all $i \ne j$.
It follows that $\map {\sigma_1} \alpha, \dots, \map {\sigma_n} \alpha$ are distinct,
so there are at least $n$ embeddings of $F \sqbrk \alpha$ into $\overline{F}$ fixing $F$,
so
:$\index {F \sqbrk \alpha} F_s \ge n$
but since $F \sqbrk \alpha / F$ is a subextension of $E / F$, by Subextensions of Separable Field Extension are Separable, $F \sqbrk \alpha / F$ is separable,
so
:$\index {F \sqbrk \alpha} F = \index {F \sqbrk \alpha} F_s \ge n$
On the other hand, by Tower Law,
:$\index {F \sqbrk \alpha} F \le \index E F = n$
Hence
:$\index {F \sqbrk \alpha} F = \index E F = n$
by Tower Law, this implies
:$\index E {F \sqbrk \alpha} = 1$
we conclude that $E = F \sqbrk \alpha$.
== Also see ==
* Steinitz's Theorem
== Sources ==
* : Chapter $1$: Galois Theory: $\S 5$: The Fundamental Theorem of Galois Theory: Corollary $5.7$
* : Chapter $\text V$: $\S 4$: Separable Extensions: Theorem $4.6$
* 
Category:Field Extensions
Category:Named Theorems
Primitive Element of Cyclic Modulo Group/Examples/Modulo 7

== Examples of Primitive Elements of Cyclic Modulo Group ==
Consider the multiplicative group of reduced residues $\Z'_p$.
The elements $\eqclass 3 7$ and $\eqclass 5 7$ are the primitive elements of $\Z'_p$.
== Proof ==
From Multiplicative Group of Reduced Residues Modulo 7 is Cyclic it is demonstrated that $\eqclass 3 7$ is indeed a generator of $\Z'_p$.
It is also shown in passing that $\eqclass 5 7$ is the inverse of $\eqclass 3 7$.
It follows from Inverse of Generator of Cyclic Group is Generator that $\eqclass 5 7$ is therefore also a generator of $\Z'_p$.
== Sources ==
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Category:Examples of Primitive Elements of Cyclic Modulo Groups