$a + bp^\frac{1}{3} + cp^\frac{2}{3} = 0$

There's a typo in the first step. @mjw caught it. Note that you're applying the quadratic formula on $p^{1/3}$, so $$p^{1/3} = \dfrac{-b ± \sqrt{b^2 - 4ac}}{2c}, \text{ if }c \ne 0.$$ Another missing link in your proof is your lack of argument about the claimed irrationality of $± \sqrt{b^2 - 4ac} - 2cp^{2/3}$.

As you handled the degenerate case "$c = 0$ and $b \ne 0$" well, we'll keep the assumption "$c \ne 0$ or $b = 0$" for the rest of the proof. Also we assume that $\sqrt{b^2-4ac}$ is irrational. Multiply $2c$ on both side of the above equality, then cube it.

\begin{align} 8c^3p =& -b^3 \pm 3b^2 \sqrt{b^2-4ac} - 3b(b^2-4ac) \pm (b^2-4ac)^{3/2} \\ =& -4b^3+12abc \pm 4(b^2-ac) \sqrt{b^2-4ac} \\ 2c^3p =& -b^3 + 3abc \pm (b^2-ac) \sqrt{b^2-4ac} \end{align}

Make $\sqrt{b^2-4ac}$ the subject of the above equality. If $b^2-ac \ne 0$,

$$\sqrt{b^2-4ac} = \pm \frac{2c^3p + b^3 - 3abc}{b^2-ac} \in \mathbb{Q},$$ contradicting our assumption on the irrationality of $\sqrt{b^2-4ac}$ if $b^2 - ac \ne 0$.

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Assume that $b^2 - ac = 0$. Then $b^2 - 4ac = b^2 = -3b^2$, and $$p^{1/3} = \frac{(-b\pm\sqrt3|b|i)}{2c} = \begin{cases} \frac{b}{c} e^{2\pi i/3} \text{ or } \frac{b}{c} e^{4\pi i/3} \text{ if } b > 0 \\ \frac{b}{c} e^{\pi i/3} \text{ or } \frac{b}{c} e^{5\pi i/3} \text{ if } b < 0 \end{cases}.$$

In this case, if $b \ne 0$, we don't have the desired conclusion, since $p = \left(\dfrac{b}{c}\right)^3$ might not be an integer, so it's not a perfect cube.

If $b = 0$, we use the assumption $b^2 = ac$, we've $a = 0$ or $c = 0$.

• If $a = 0$, only the term $cp^{2/3} = 0$ is left in the original equation, but $p \ne 0$ as $p$ can't be a perfect cube, so $c = 0$.
• If $c = 0$, the original equation becomes $a = 0$. Done.

I have somewhat weird way to see this. Consider the system \begin{align*} a + b p^{1/3} + c p^{2/3} & = 0\\ cp + a p^{1/3} + b p^{2/3} & = 0\\ bp + cp p^{1/3} + a p^{2/3} & = 0 \end{align*} or $$\begin{pmatrix} a & b & c \\ cp & a & b \\ bp & cp & a \end{pmatrix} \begin{pmatrix} 1 \\ p^{1/3} \\ p^{2/3} \end{pmatrix} =0. $$

So the coefficient matrix has zero determinant, i.e. $$a^3 + b(b^2-3ac)p + c^3 p^2=0.$$ Now we can proceed with infinite descent.

(The essence is until here, and below is just some calculations.)

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Note that we can assume that $p$ is an integer; If $a + b (n/d)^{1/3} + c (n/d)^{2/3}=0$ then $$ad^2 + bd\cdot d^{2/3} n^{1/3} + c d^{4/3}n^{2/3}=0,$$ i.e. $$ad^2 + bd\cdot (d^2n)^{1/3} + c (d^2n)^{2/3}=0$$ so we are reduced to the integer $p$ case.

Let $q$ be a prime factor or $p$. One can assume that $q^3 \not\mid p$; in this case $q$ factor is absorbed into coefficients $b$ and $c$. Assume $(a, b, c)$ is a nontrivial integer solution. We have two cases;

• Case 1: Let $p = qN$ with $q\not\mid N$. Then $$a^3 + b(b^2-3ac)qN + c^3 q^2N^2=0,$$ i.e. $a = qA$. Then $$q^2A^3 + b(b^2-3qAc)N + c^3 qN^2=0,$$ i.e. $b = qB$. Then again $$qA^3 + B(q B^2-3Ac)qN + c^3 N^2=0,$$ i.e. $q|c$, i.e. $c = qC$, and $$A^3 + B(B^2-3AC)qN + C^3 q^2 N^2 = A^3 + B(B^2-3AC)p + C^3 p^2 =0.$$ Thus, if $(a, b, c)$ is an integer solution then $(a/q, b/q, c/q)$ also is an integer solution; this descent cannot be done infinitely since $a, b, c$ are finite, i.e. a contradiction.

• Case 2 : Let $p = q^2 N $ with $q\not\mid N$. Then $$a^3 + b(b^2-3ac)q^2 N + c^3 q^4 N^2=0.$$ One can assert $a = q A$, then $$q A^3 + b(b^2-3qAc) N + q^2c^3 N^2=0,\quad \mathbf{(**)}$$ i.e. $b = qB$. Thus $$ A^3 + B(q B^2-3Ac)q N + q c^3 N^2=0,$$ i.e. $A$ can be divided by $q$ once again. Let $A = qA'$ to have $$ q^2 A'^3 + B( B^2-3A'c)q N + c^3 N^2=0,$$ i.e. $c = qC$, $$ q A'^3 + B( B^2-3qA'C) N + q^2 C^3 N^2=0 \quad \mathbf{(**)}$$ Compare two equations marked by (**); If $(A, b, c)$ satisfies $$q A^3 + b(b^2-3qAc) N + q^2c^3 N^2=0 $$ then we have another integer solution $(A/q, b/q, c/q)$. So, again by infinite descent, there is no such $(a, b, c)$.

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This method also works for $p^{1/4}$ case.

I think I have never seen the matrix of the form $$\begin{pmatrix} a & b & c \\ cp & a & b \\ bp & cp & a \end{pmatrix} $$ or its variants. Are there any reference?

Let $K= \mathbb{Q}(p^{\frac{1}{3}}) \cong \mathbb{Q}[x]/(x^3-p).$ We have the following:

1. $\text{Tr}_{K/\mathbb{Q}} (p^{\frac{1}{3}}) = 0 $, as the minimal polynomial of $p^{\frac{1}{3}}$ is $x^3-p.$
2. $\text{Tr}_{K/\mathbb{Q}} (p^{\frac{2}{3}}) = 0 $, as the minimal polynomial of $p^{\frac{2}{3}}$ is $x^3-p^2.$
3. $\text{Tr}_{K/\mathbb{Q}} (a) = 3a$, for all $a \in \mathbb{Q}.$

Now applying the trace to your equation we get \begin{align*} \text{Tr}_{K/\mathbb{Q}}(a + bp^\frac{1}{3} + cp^\frac{2}{3}) &= \text{Tr}_{K/\mathbb{Q}}(a)+ b \text{Tr}_{K/\mathbb{Q}}(p^\frac{1}{3} ) + c \text{Tr}_{K/\mathbb{Q}}(p^\frac{2}{3})\\ &= \text{Tr}_{K/\mathbb{Q}}(a)+0+0 = 3a = 0= \text{Tr}_{K/\mathbb{Q}}(0),\\ \end{align*} thus $a=0.$ Next multiply your equation by $p^{\frac{1}{3}}$, apply the trace, and conclude that $c=0.$ Repeat.

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Edit: As Paramanand Singh correctly points out the problem reduces to showing that $f(x)= x^3-p$ is the minimal polynomial of $p^{\frac{1}{3}},$ which I assumed. However, this follows directly from Eisenstein's Criterion and the fact that $f(p^{\frac{1}{3}})=0.$ In light of this information, the polynomial given by the OP is of degree $2$, thus must be the zero polynomial.