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Solve the following homogeneous systems of linear equations with coefficients over field $\mathbb{F}_2$:

 x1+x2+x3+x4+x5+x6+x7 = 0
 x1=0
 x2+x3+x4 =0
 x5=0

I want to output on the sage following:

x1=x5=0
x6=x7
x2+x3+x4=0

I want to solve a general homogeneous systems with coefficients over field $\mathbb{F}_2$ and output same as above.

I hope that someone can help. Thanks!

My code in sagemath

    def bruteforce
    count = 0
     s = [0, 1, 2, 4, 8]
   s.each do |a|
s.each do |b|
  s.each do |c|
    s.each do |d|
      s.each do |e|
        if a+b+c+d+e == 8
          count += 1
          puts "#{count}.: #{a}+#{b}+#{c}+#{d}+#{e}"
          end
        end
      end
     end
    end
   end
  count
  end

but it was an error as below

  File "<ipython-input-1-87370adfacc9>", line 1
       def brutefor 
    SyntaxError: invalid syntax

Find all non-negative integer a, b, c,d, e such that

$$a+b+c+d+e = 8$$

Is there any method for this? I have no idea. I can just fix the limit.

I use the code above but it error.

File "<ipython-input-1-b39e5d3b522e>", line 3 Multivariate Polynomial Ring in x1, x2, x3, x4 over Finite Field of size Integer(2) SyntaxError: invalid syntax

Let $P_k:= \mathbb{F}_2[x_1,x_2,\ldots ,x_k]$ be the polynomial algebra in $k$ variables with the degree of each $x_i$ being $1,$ regarded as a module over the mod-$2$ Steenrod algebra $\mathcal{A}.$ Here $\mathcal{A} = \langle Sq^{2^m}\,\,|\,\,m\geq 0\rangle.$

Being the cohomology of a space, $P_k$ is a module over the mod-2 Steenrod algebra $\mathscr{A}.$ The action of $\mathscr{A}$ on $P_k$ is explicitly given by the formula

$$Sq^m(x_j^d) = \binom{d}{m}x_j^{m+d},$$ where $ \binom{d}{m}$ is reduced mod-2 and $\binom{d}{m} = 0$ if $m > d.$

Now, I want to use the Steenrod algebra package and Multi Polynomial ring package and using formular above to construction of formula following in Sagemath program

$$ Sq^m(f) = \sum\limits_{2^{m_1} + 2^{m_2} + \cdots + 2^{m_k}= m}\binom{d_1}{2^{m_1}}x_1^{2^{m_1}+d_1}\binom{d_1}{2^{m_2}}x_2^{2^{m_2}+d_2}\ldots \binom{d_k}{2^{m_k}}x_k^{2^{m_k}+d_k}.$$ forall $f = x_1^{d_1}x_2^{d_2}\ldots x_k^{d_k}\in P_k$

Example: Let $k = 5, m = 2$ and $f = x_1^2x_2^3x_3^2x_4x_5\in P_5.$ We have $$ Sq^2(x_1^2x_2^3x_3^2x_4x_5) = x_1^4x_2^3x_3^2x_4x_5 + x_1^2x_2^5x_3^2x_4x_5 + x_1^2x_2^3x_3^4x_4x_5 +x_1^2x_2^3x_3^2x_4^2x_5^2 + x_1^2x_2^4x_3^2x_4x_5^2 + x_1^2x_2^4x_3^2x_4^2x_5^1.$$

I hope that someone can help. Thanks!