Rendered TeX expressions in df7e0a3

README.md

@@ -21,21 +21,21 @@ A **Galois field** <img src="/tex/e749ad7d18f2855210ea115451c98828.svg?invert_in
 
 ### Prime fields
 
-Any Galois field has a unique *characteristic* p, the minimum positive p such that p(1) = 1 + ... + 1 = 0, and p is prime. The smallest Galois field of characteristic p is a **prime field**, and any Galois field of characteristic p is a *finite-dimensional vector space* over its prime subfield.
+Any Galois field has a unique *characteristic* <img src="/tex/2ec6e630f199f589a2402fdf3e0289d5.svg?invert_in_darkmode&sanitize=true" align=middle width=8.270567249999992pt height=14.15524440000002pt/>, the minimum positive <img src="/tex/2ec6e630f199f589a2402fdf3e0289d5.svg?invert_in_darkmode&sanitize=true" align=middle width=8.270567249999992pt height=14.15524440000002pt/> such that <img src="/tex/c89470a5c4f7e9473c0ebeb6148bdf5d.svg?invert_in_darkmode&sanitize=true" align=middle width=157.12847205pt height=24.65753399999998pt/>, and <img src="/tex/2ec6e630f199f589a2402fdf3e0289d5.svg?invert_in_darkmode&sanitize=true" align=middle width=8.270567249999992pt height=14.15524440000002pt/> is prime. The smallest Galois field of characteristic <img src="/tex/2ec6e630f199f589a2402fdf3e0289d5.svg?invert_in_darkmode&sanitize=true" align=middle width=8.270567249999992pt height=14.15524440000002pt/> is a **prime field**, and any Galois field of characteristic <img src="/tex/2ec6e630f199f589a2402fdf3e0289d5.svg?invert_in_darkmode&sanitize=true" align=middle width=8.270567249999992pt height=14.15524440000002pt/> is a *finite-dimensional vector space* over its prime subfield.
 
-For example, GF(4) is a Galois field of characteristic 2 that is a two-dimensional vector space over the prime subfield GF(2) = Z / 2Z.
+For example, <img src="/tex/b566101ab803ed6b496286a86485ba7d.svg?invert_in_darkmode&sanitize=true" align=middle width=44.634829799999984pt height=24.65753399999998pt/> is a Galois field of characteristic 2 that is a two-dimensional vector space over the prime subfield <img src="/tex/b5dc0ba9c96f491c1071e58918af956a.svg?invert_in_darkmode&sanitize=true" align=middle width=107.05713974999998pt height=24.65753399999998pt/>.
 
 ### Extension fields
 
-Any Galois field has order a prime power p^q for prime p and positive q, and there is a Galois field GF(p^q) of any prime power order p^q that is *unique up to non-unique isomorphism*. Any Galois field GF(p^q) can be constructed as an **extension field** over a smaller Galois subfield GF(p^r), through the identification GF(p^q) = GF(p^r)[X] / \<f(X)\> for an *irreducible monic polynomial* f(X) of degree q - r + 1 in the *polynomial ring* GF(p^r)[X].
+Any Galois field has order a prime power <img src="/tex/f5bb40e395f7fadf8da15b1de7cb359e.svg?invert_in_darkmode&sanitize=true" align=middle width=14.70840524999999pt height=21.839370299999988pt/> for prime <img src="/tex/2ec6e630f199f589a2402fdf3e0289d5.svg?invert_in_darkmode&sanitize=true" align=middle width=8.270567249999992pt height=14.15524440000002pt/> and positive <img src="/tex/d5c18a8ca1894fd3a7d25f242cbe8890.svg?invert_in_darkmode&sanitize=true" align=middle width=7.928106449999989pt height=14.15524440000002pt/>, and there is a Galois field <img src="/tex/e749ad7d18f2855210ea115451c98828.svg?invert_in_darkmode&sanitize=true" align=middle width=51.94591709999999pt height=24.65753399999998pt/> of any prime power order <img src="/tex/f5bb40e395f7fadf8da15b1de7cb359e.svg?invert_in_darkmode&sanitize=true" align=middle width=14.70840524999999pt height=21.839370299999988pt/> that is *unique up to non-unique isomorphism*. Any Galois field <img src="/tex/e93021dfa6f70beff0eeaac3502698a8.svg?invert_in_darkmode&sanitize=true" align=middle width=54.09429629999998pt height=24.65753399999998pt/> can be constructed as an **extension field** over a smaller Galois subfield <img src="/tex/7f17ac2b8db44db36d376e44dbca459b.svg?invert_in_darkmode&sanitize=true" align=middle width=51.96550754999999pt height=24.65753399999998pt/>, through the identification <img src="/tex/676c07f4913e5983d30644120a2190a0.svg?invert_in_darkmode&sanitize=true" align=middle width=208.38632594999999pt height=24.65753399999998pt/> for an *irreducible monic polynomial* <img src="/tex/161805ece9a8142e4ebe9d356fd0f763.svg?invert_in_darkmode&sanitize=true" align=middle width=37.51151249999999pt height=24.65753399999998pt/> of degree <img src="/tex/610a0cab14ccd2bc1d2080214ad87a19.svg?invert_in_darkmode&sanitize=true" align=middle width=64.20263354999999pt height=21.18721440000001pt/> in the *polynomial ring* <img src="/tex/16fd46b712782718e9e17184e9d36aa0.svg?invert_in_darkmode&sanitize=true" align=middle width=76.00662299999999pt height=24.65753399999998pt/>.
 
-For example, GF(4) has order 2^2 and can be constructed as an extension field GF(2)[X] / \<f(X)\> where f(X) = X^2 + X + 1 is an irreducible monic quadratic polynomial in GF(2)[X].
+For example, <img src="/tex/b566101ab803ed6b496286a86485ba7d.svg?invert_in_darkmode&sanitize=true" align=middle width=44.634829799999984pt height=24.65753399999998pt/> has order <img src="/tex/ec4089d7f3fb410f521723b967e41a69.svg?invert_in_darkmode&sanitize=true" align=middle width=14.771756999999988pt height=26.76175259999998pt/> and can be constructed as an extension field <img src="/tex/eff7dfe6e7384ba3c57607f1376fda7b.svg?invert_in_darkmode&sanitize=true" align=middle width=127.19210129999998pt height=24.65753399999998pt/> where <img src="/tex/14204002b85c3f046fbde37b519e83c8.svg?invert_in_darkmode&sanitize=true" align=middle width=145.02252929999997pt height=26.76175259999998pt/> is an irreducible monic quadratic polynomial in <img src="/tex/b5b165971fad7e6b6c890db109b9d2fd.svg?invert_in_darkmode&sanitize=true" align=middle width=68.67594524999998pt height=24.65753399999998pt/>.
 
 ### Binary fields
 
-A Galois field of the form GF(2^m) for big positive m is a sum of X^n for a non-empty set of 0 \< n \< m. For computational efficiency in cryptography, an element of a **binary field** can be represented by an integer that represents a bit string. It should always be used when the field characteristic is 2.
+A Galois field of the form <img src="/tex/f4687471921caacf38fd0e0667005c1f.svg?invert_in_darkmode&sanitize=true" align=middle width=57.12159419999999pt height=24.65753399999998pt/> for big positive <img src="/tex/0e51a2dede42189d77627c4d742822c3.svg?invert_in_darkmode&sanitize=true" align=middle width=14.433101099999991pt height=14.15524440000002pt/> is a sum of <img src="/tex/aedfd2f0682e37eedb201bcd2ca04442.svg?invert_in_darkmode&sanitize=true" align=middle width=23.034689699999987pt height=22.465723500000017pt/> for a non-empty set of <img src="/tex/ddce1c5958e2d066abe368e96fa697db.svg?invert_in_darkmode&sanitize=true" align=middle width=76.35444795pt height=21.18721440000001pt/>. For computational efficiency in cryptography, an element of a **binary field** can be represented by an integer that represents a bit string. It should always be used when the field characteristic is 2.
 
-For example, X^8 + X^4 + X^3 + X + 1 can be represented as the integer 283 that represents the bit string 100011011.
+For example, <img src="/tex/72891ad284e66d0f200979d5f8209d4e.svg?invert_in_darkmode&sanitize=true" align=middle width=170.34202019999998pt height=26.76175259999998pt/> can be represented as the integer 283 that represents the bit string 100011011.
 
 ## Example usage