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Eilenberg–Niven theorem generalizes the fundamental theorem of algebra for quaternionic polynomials,¹ and this theorem was further generalized to also encompass octonionic polynomials.²

Does similar theorem holds for the sedenion algebra?

References:

[1] Eilenberg, Samuel; Niven, Ivan, The “fundamental theorem of algebra” for quaternions, Bull. Am. Math. Soc. 50, 246-248 (1944). ZBL0063.01228.

[2] Jou, Yuh-Lin, The ’fundamental theorem of algebra’ for Cayley numbers, Sci. Record, Acad. Sinica 3, 29-33 (1950). ZBL0039.26701.

Edit: This question is indeed answered in the negative (see the comment of user49822). Of course the sedenion algebra has zero divisors, f.i. $a = e_1+e_{10}$. But each $x \in \mathbb{S} \setminus \{0\}$ has an inverse $1/x = \bar{x}/\|x\|^2$ with $x \cdot 1/x = 1/x \cdot x = 1$. In particular $1/a = -a/2$.

The Eilenberg–Niven theorem generalizes the fundamental theorem of algebra for quaternionic polynomials,¹ and this theorem was further generalized to also encompass octonionic polynomials.²

Does similar theorem holds for the sedenion algebra?

References:

[1] Eilenberg, Samuel; Niven, Ivan, The “fundamental theorem of algebra” for quaternions, Bull. Am. Math. Soc. 50, 246–248 (1944). ZBL0063.01228.

[2] Jou, Yuh-Lin, The ‘fundamental theorem of algebra’ for Cayley numbers, Sci. Record, Acad. Sinica 3, 29–33 (1950). ZBL0039.26701.

Edit: This question is indeed answered in the negative (see the comment of user49822). Of course the sedenion algebra has zero divisors, f.i. $a = e_1+e_{10}$. But each $x \in \mathbb{S} \setminus \{0\}$ has an inverse $1/x = \bar{x}/\|x\|^2$ with $x \cdot 1/x = 1/x \cdot x = 1$. In particular $1/a = -a/2$.

Eilenberg–Niven theorem generalizes the fundamental theorem of algebra for quaternionic polynomials,¹ and this theorem was further generalized to also encompass octonionic polynomials.²

Does similar theorem holds for the sedenion algebra?

References:

[1] Eilenberg, Samuel; Niven, Ivan, The “fundamental theorem of algebra” for quaternions, Bull. Am. Math. Soc. 50, 246-248 (1944). ZBL0063.01228.

[2] Jou, Yuh-Lin, The ’fundamental theorem of algebra’ for Cayley numbers, Sci. Record, Acad. Sinica 3, 29-33 (1950). ZBL0039.26701.

Edit: This question is still open. Of course the sedenion algebra has zero divisors, f.i. $a = e_1+e_{10}$. But each $x \in \mathbb{S} \setminus \{0\}$ has an inverse $1/x = \bar{x}/\|x\|^2$ with $x \cdot 1/x = 1/x \cdot x = 1$. In particular $1/a = -a/2$.

Eilenberg–Niven theorem generalizes the fundamental theorem of algebra for quaternionic polynomials,¹ and this theorem was further generalized to also encompass octonionic polynomials.²

Does similar theorem holds for the sedenion algebra?

References:

[1] Eilenberg, Samuel; Niven, Ivan, The “fundamental theorem of algebra” for quaternions, Bull. Am. Math. Soc. 50, 246-248 (1944). ZBL0063.01228.

[2] Jou, Yuh-Lin, The ’fundamental theorem of algebra’ for Cayley numbers, Sci. Record, Acad. Sinica 3, 29-33 (1950). ZBL0039.26701.

Edit: This question is indeed answered in the negative (see the comment of user49822). Of course the sedenion algebra has zero divisors, f.i. $a = e_1+e_{10}$. But each $x \in \mathbb{S} \setminus \{0\}$ has an inverse $1/x = \bar{x}/\|x\|^2$ with $x \cdot 1/x = 1/x \cdot x = 1$. In particular $1/a = -a/2$.

Eilenberg–Niven theorem generalizes the fundamental theorem of algebra for quaternionic polynomials,¹ and this theorem was further generalized to also encompass octonionic polynomials.²

Does similar theorem holds for the sedenion algebra?

References:

[1] Eilenberg, Samuel; Niven, Ivan, The “fundamental theorem of algebra” for quaternions, Bull. Am. Math. Soc. 50, 246-248 (1944). ZBL0063.01228.

[2] Jou, Yuh-Lin, The ’fundamental theorem of algebra’ for Cayley numbers, Sci. Record, Acad. Sinica 3, 29-33 (1950). ZBL0039.26701.

Eilenberg–Niven theorem generalizes the fundamental theorem of algebra for quaternionic polynomials,¹ and this theorem was further generalized to also encompass octonionic polynomials.²

Does similar theorem holds for the sedenion algebra?

References:

[1] Eilenberg, Samuel; Niven, Ivan, The “fundamental theorem of algebra” for quaternions, Bull. Am. Math. Soc. 50, 246-248 (1944). ZBL0063.01228.

[2] Jou, Yuh-Lin, The ’fundamental theorem of algebra’ for Cayley numbers, Sci. Record, Acad. Sinica 3, 29-33 (1950). ZBL0039.26701.

Edit: This question is still open. Of course the sedenion algebra has zero divisors, f.i. $a = e_1+e_{10}$. But each $x \in \mathbb{S} \setminus \{0\}$ has an inverse $1/x = \bar{x}/\|x\|^2$ with $x \cdot 1/x = 1/x \cdot x = 1$. In particular $1/a = -a/2$.