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Problem 4 Let us endow the finite set $Z_6 := \{0, 1, 2, 3, 4, 5\}$ with the operations and $+: Z_6 \times Z_6 \to Z_6$ $(x, y) \to r_6(x + y)$ $\cdot: Z_6 \times Z_6 \to Z_6$ $(x, y) \to r_6(x \cdot y)$ where, for an integer $a$, $r_6(a)$ is the remainder in the Euclidean division of $a$ by 6 i.e. the unique integer $0 \le r_6(a) < 6$ such that $a = 6 \times q + r_6(a)$. Show that $Z_6$ fails to satisfy all the axioms for a field. (Hint: look at axioms 7 and 8. What is the multiplicative inverse of 2?)

          Problem 4 Let us endow the finite set $Z_6 := \{0, 1, 2, 3, 4, 5\}$ with the operations

and
$+: Z_6 \times Z_6 \to Z_6$
$(x, y) \to r_6(x + y)$
$\cdot: Z_6 \times Z_6 \to Z_6$
$(x, y) \to r_6(x \cdot y)$
where, for an integer $a$, $r_6(a)$ is the remainder in the Euclidean division of $a$ by 6 i.e. the
unique integer $0 \le r_6(a) < 6$ such that $a = 6 \times q + r_6(a)$.
Show that $Z_6$ fails to satisfy all the axioms for a field. (Hint: look at axioms 7 and 8.
What is the multiplicative inverse of 2?)

Problem 4 Let us endow the finite set Z6 := {0, 1, 2, 3, 4, 5} with the operations

and
+: Z6 × Z6 → Z6
(x, y) → r6(x + y)
·: Z6 × Z6 → Z6
(x, y) → r6(x · y)
where, for an integer a, r6(a) is the remainder in the Euclidean division of a by 6 i.e. the
unique integer 0 ≤ r6(a) < 6 such that a = 6 × q + r6(a).
Show that Z6 fails to satisfy all the axioms for a field. (Hint: look at axioms 7 and 8.
What is the multiplicative inverse of 2?)

Added by Tanner P.

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