MATH 470 Lecture 24

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Recall

• $\mathbb {F} _{p}$, where $p$ is prime, represents a prime field
• $\mathbb {F} _{q}$, where $q$ is a prime power not a prime, represents a finite extension field.

Consider $\mathbb {F} _{27}$ realized as $\mathbb {F} _{3}\left[T\right]/\left\langle T^{3}+2T+1\right\rangle$.

We need to check that $T^{3}+2T+1$ is irreducible (i.e. it has no non-trivial roots)

If it had a non-trivial factorization, it would be (linear) × (quadratic)

$t$	$t^{3}+2t+1$
0	1
1	1
2	1

Therefore there are no roots mod 3.

Therefore, $\mathbb {F} _{3}\left[T\right]/\left\langle T^{3}+2T+1\right\rangle$ is a realization of $\mathbb {F} _{3^{3}}=\mathbb {F} _{27}$

Inverses

What is ${\frac {1}{T+1}}$ in this realization of $\mathbb {F} _{27}$?

It would be the polynomial $p(T)\in \mathbb {F} _{3}\left[T\right]$ with $p(T)\,(T+1)=1$.

This is equivalent to finding polynomials $p(T),q(T)\in \mathbb {F} _{3}\left[T\right]$ with $p(T)\,(T+1)+q(T)\,(T^{3}+2T+1)=1$.

When working with integers, we had the extended Euclidean algorithm. We can still use it, but we need a few modifications:

${\begin{bmatrix}1&0\\0&1\\a(T)&b(T)\end{bmatrix}}\to {\begin{bmatrix}*(T)&u(T)\\*(T)&v(T)\\0&1\end{bmatrix}}$

Our goal is to make $a$ and $b$ smaller in degree (rather than absolute value, as we did with integers).

Example

To compute ${\frac {1}{T+1}}$ in the above realization of $\mathbb {F} _{27}$, we do...

${\begin{bmatrix}1&0\\0&1\\T+1&T^{3}+2T+1\end{bmatrix}}\to {\begin{bmatrix}1&-T^{2}\\0&1\\T+1&-T^{2}+2T+1\end{bmatrix}}\to {\begin{bmatrix}1&-T^{2}+T\\0&1\\T+1&3T+1\end{bmatrix}}\to {\begin{bmatrix}1&-T^{2}+T-3\\0&1\\T+1&-2\end{bmatrix}}\to {\begin{bmatrix}1&2T^{2}+T\\0&1\\T+1&1\end{bmatrix}}$

Therefore, $(2T^{2}+T)\,(T+1)+(1)\,(T^{3}+2T+1)=1$, and ${\frac {1}{T+1}}=2T^{2}+T$ in this $\mathbb {F} _{27}$.

Note that there are possibly many distinct realizations of $\mathbb {F} _{q}$: this is why the irreducible polynomial must be clarified!

Is $T^{3}+2T+1$ primitive over $\mathbb {F} _{3}$? (i.e. do the powers $1,T,T^{2},\ldots ,T^{26}=\mathbb {F} _{27}^{*}$?

What do we know about the order of $T$ in $\mathbb {F} _{27}^{*}$.

Recall: the order of T in a field $\mathbb {F} _{q}$ would be the smallest $k$ with $T^{k}=1$ in $\mathbb {F} _{q}$

Lagrange's Theorem

Implies that the order of any $T\in \mathbb {F} _{q}^{*}$ must divide the cardinality (number of elements) of $\mathbb {F} _{q}^{*}$.

The cardinality of $\mathbb {F} _{q}^{*}$ is $q-1$, so ${\text{order}}\mid (q-1)$

For our realization of $\mathbb {F} _{27}$, the order could be 1, 2, 13, or 26

• order of $T$ is not 1 since $T\neq 1$ in this realization.
• order of $T$ is not 2 since all 27 elements in this realization are distinct
• order of $T$ is not 13 since $T^{13}=2$
• Therefore, by elimination, the order of T is 26

Therefore, $T^{3}+2T+1$ is primitive over $\mathbb {F} _{3}$:

Elements of $\mathbb {F} _{27}$ realized by $\mathbb {F} _{3}[T]/\left\langle T^{3}+2T+1\right\rangle$

$n$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^n}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^n \pmod{T^3+2T+1} \pmod{3}}
0	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}
1	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}	$T$
2	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^2}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^2}
3	$T^{3}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2T-1 \equiv T+2}
4	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^4}	$T^{2}+2T$
5	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^5}	$(T+2)+2T^{2}\equiv 2T^{2}+T+2$
6	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^6}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (T+2) \, (T+2) \equiv T^2 + T + 1}
7	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^7}	$(T+2)+T^{2}+T\equiv T^{2}+2T+2$
8	$T^{8}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (T+2) + 2T^2 + 2T \equiv 2T^2 + 2}
9	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^9}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (T+2)^3 \equiv (T+2) + 2 \equiv T + 1}
10	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{10}}	$T^{2}+T$
11	$T^{11}$	$(T+2)+T^{2}\equiv T^{2}+T+2$
12	$T^{12}$	$(T+2)+T^{2}+2T\equiv T^{2}+2$
13	$T^{13}$	$(T+2)+2T\equiv 2$
14	$T^{14}$	$2T$
15	$T^{15}$	$2T^{2}$
16	$T^{16}$	$2(T+2)\equiv 2T+1$
17	$T^{17}$	$2T^{2}+T$
18	$T^{18}$	$2(T+2)+T^{2}\equiv T^{2}+2T+1$
19	$T^{19}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (T+2)+2T^2+T \equiv 2T^2 + 2T + 2}
20	$T^{20}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2(T+2) + 2T^2 + 2T \equiv 2T^2 + T + 1}
21	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{21}}	$2(T+2)+T^{2}+T\equiv T^{2}+1$
22	$T^{22}$	$(T+2)+T\equiv 2T+2$
23	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{23}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2T^2+2T}
24	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{24}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2(T+2) + 2T^2 \equiv 2T^2 + 2T + 1}
25	$T^{25}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2(T+2) + 2T^2 + T \equiv 2T^2 + 1}
26	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{26}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2(T+2) + T \equiv 1}

In sage,

k = GF(3^3, 'T') for i,x in enumerate(k): print i,x

Elliptic Curves

Consider elliptic curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E : y^2 = x^3 + x + 1} over $\mathbb {F} _{q}$ realized as $\mathbb {F} _{3}\left[9\right]/\left\langle T^{2}-2\equiv T^{2}+1\right\rangle$

First: what are the squares in (this) $\mathbb {F} _{9}$?

$x$ or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3+x+1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
$1$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}	$2$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} *	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T+1}	$2T$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T+2}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}
$2T$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}	$1$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2T+1}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}	$0$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2T+2}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2T}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}

* Note: 2 has no square root mod 3, but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2} = \pm{T}} in this $\mathbb {F} _{9}$

The points in this Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_9} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} are:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0, \pm 1)}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1, 0)}
• $(2,\pm T)$
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (T+1, 0)}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (T+2, \pm T)}
• $(2T,\pm 1)$
• $(2T+1,0)$
• $(2T+2,\pm T)$

There is also the point at infinity for a grand total of 16 points.

Hasse's Theorem

(1933)

If $E$ is any elliptic curve, over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_q} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} points, then

$\left|n-(q+1)\right|\leq 2{\sqrt {q}}$

(conjectured by Emil Artin in his 1924 Ph.D. thesis, and ultimately led to P. Deligne winning a fields medal in 1974 for the riemann hypothesis for function fields)

This comes from the strange connection between finite fields and complex numbers in Genus 1

For our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} , we get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 16} , so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left|16 - (9+1)\right| &\le 2\sqrt{9} \\ \left|6\right| &\le 2 \cdot 3 \end{align}}

And this checks out.

Furthermore, Hasse's theorem implies that over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_9} , the number of points must be between 4 and 16.

Addition Law

How does the addition law work here? (e.g. $(1,0)\oplus (2,T)=\Box$ ?)

Recall:

When adding two points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y) \oplus (x',y') = (\mathbb{X}, \mathbb{Y})} ,

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = \begin{cases} \frac{3x^2+1}{2y} & x = x' \\ \frac{y'-y}{x'-x}\end{cases}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{X} = m^2 - x - x'} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Y} = m(x-\mathbb{X}) - y}

So going back to our example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,0) \oplus (2,T)} , we get

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = \frac{T-0}{2-1} = T}
• $\mathbb {X} =T^{2}-1-2=T^{2}=2$
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Y} = m(x-\mathbb{X}) - y = 2T}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,0) \oplus (2,T) = (2, 2T)}